Method, apparatus and medium for decoding

ABSTRACT

A method of utilizing an N-point primary transform core matrix to encode may include determining whether to use a first transform core matrix that is an N-point 8-bit primary transform core matrix. When a result of the determination is to use the first transform core matrix, the method may include: (i) constructing the first transform core based on a set of unique numbers, and (ii) encoding using the derived primary transform core. The method may further include, when a result of the determination is to not use the first primary transform: (i) deriving a second transform core based on the same set of unique numbers, and (ii) encoding using the derived second transform core.

CROSS-REFERENCE TO RELATED APPLICATION(S)

This is a continuation application of U.S. Non-Provisional applicationSer. No. 17/180,096, filed Feb. 19, 2021, which is a continuationapplication of U.S. Non-Provisional application Ser. No. 16/821,669,filed Mar. 17, 2020, which is a continuation application of U.S.Non-Provisional application Ser. No. 16/235,995, filed on Dec. 28, 2018,which claims the benefit of priority of U.S. Provisional Application No.62/692,658, filed on Jun. 29, 2018, in the United States Patent &Trademark Office, the disclosures of which are herein incorporated byreference in their entireties.

BACKGROUND Field

This disclosure is directed towards video compression technologies. Inparticular, the present disclosure is directed towards next-generationvideo coding technologies including video coding/decoding technologiesbeyond High Efficiency Video Coding (HEVC), such as Versatile VideoCoding (VVC). More specifically, an aspect of the disclosure is directedtowards a VVC primary transform method, device and computer medium thatuse a large, 8-bit transform core having a size that is equal to orgreater than 64×64.

Description of Related Art

Video coding and decoding using inter-picture prediction with motioncompensation has been known for decades. Uncompressed digital video canconsist of a series of pictures, each picture having a spatial dimensionof, for example, 1920×1080 luminance samples and associated chrominancesamples. The series of pictures can have a fixed or variable picturerate (informally also known as frame rate), of, for example 60 picturesper second or 60 hertz (Hz). Uncompressed video has significant bitraterequirements. For example, 1080p60 4:2:0 video at 8 bit per sample(1920×1080 luminance sample resolution at 60 Hz frame rate) requiresclose to 1.5 Gbit/s bandwidth. An hour of such video requires more than600 GByte of storage space.

One purpose of video coding and decoding can be the reduction ofredundancy in the input video signal, through compression. Compressioncan help reduce aforementioned bandwidth or storage space requirements,in some cases by two orders of magnitude or more. Both lossless andlossy compression, as well as a combination thereof can be employed.Lossless compression refers to techniques where an exact copy of theoriginal signal can be reconstructed from the compressed originalsignal. When using lossy compression, the reconstructed signal may notbe identical to the original signal, but the distortion between originaland reconstructed signal is small enough to make the reconstructedsignal useful for the intended application. In the case of video, lossycompression is widely employed. The amount of distortion tolerateddepends on the application; for example, users of certain consumerstreaming applications may tolerate higher distortion than users oftelevision contribution applications. The compression ratio achievablecan reflect that: higher allowable/tolerable distortion can yield highercompression ratios.

A person of ordinary skill generally understands videocompression/decompression technology. In general, to compress video orimage data, a series of functional steps are performed to result in acompressed video or image file. Although an image, such as a 360° image(e.g., captured by a 360° camera) may be suitable for compression, forease of reading, compression of a video file will be explained. Togenerate a compressed video file, under conventional standards (e.g.,H.264, H.265) an uncompressed video sample stream that is received froma video source may be partitioned or parsed, which results in a block ofsamples of two or more reference pictures.

Bi-Prediction can relate to techniques where a prediction unit (PU),such as a predicted block of samples, can be predicted from two motioncompensated blocks of samples of two or more reference pictures.Bi-prediction was first introduced into video coding standards inMPEG-1, and has also been included in other video coding technologiesand standards such as MPEG-2 Part 2 (or H.262), H.264 and H.265 as well.

When decompressing a compressed video file, during the reconstruction ofa sample of a bi-predicted PU, motion compensated and interpolated inputsamples from each reference block can be multiplied by a weightingfactor that can be different for each reference block, and such weightedsample values of the two reference blocks can be added to generate thesample under reconstruction. Such sample can be processed further bymechanisms such as loop filtering.

In MPEG-1 and MPEG-2, the weighting factors can be determined based onthe relative temporal distance between the picture to which the PU underreconstruction belongs to, and the two reference pictures. This ispossible because, in MPEG-1 and MPEG-2, one of the two reference I or Ppictures was in the “past”, and the other in the “future” (in terms ofpresentation order) of the B-picture under reconstruction, and becausein MPEG-1 and MPEG-2, there was a well-defined timing relationshipestablished for any picture under reconstruction in relation to itsreference pictures.

Starting with H.264, the reference picture selection concepts forbi-predicted pictures were relaxed such that the reference pictures onlyneeded to be earlier in decoding order, but not in presentation order.Further, the notion of time was also relaxed in that neither H.264 norH.265 requires a constrained/fixed picture interval in the time domain.Therefore, a decoder cannot calculate weighting factors any more basedon the timing information available in the bitstream. Instead, H.264 andH.265 include a “default” of 0.5 as the weighting factor for thereference samples of a bi-predicted picture. This default can beoverwritten by syntax available in the slice header known aspred_weight_table( ). The default of 0.5 or the information in thepred_weight_table may apply to all bi-predicted PUs in a given slice.

-   Non-Patent Literature 1 shows the H.265/HEVC standard. However, a    need for standardization of future video coding technology with a    compression capability that significantly exceeds that of the HEVC    standard (including its extensions) has been studied by the    inventors.-   Non-Patent Literature 2 discloses a recently launched    standardization format for next-generation video coding beyond HEVC    called Versatile Video Coding (VVC) with a version VTM (VVC Test    Model). VVC may generally provide a large (e.g., 64-point or higher)    transform core using a 10-bit integer matrix.

Listing of Related Art

-   Non-Patent Literature 1: “Transform design for HEVC with 16 bit    intermediate data representation”; published by: Joint Collaborative    Team on Video Coding (JCT-VC) of ITU-T SG16 WP3 and ISO/IEC    JTC1/SC29/WG11, 5th Meeting: Geneva, CH, Mar. 16-23, 2011 (Document:    JCTVC-E243). The H.265/HEVC (High Efficiency Video Coding) standard    may include Version 1 (2013), Version 2 (2014), Version 3 (2015) and    Version 4 (2016) published by ITU-T VCEG (Q6/16) ISO/IEC MPEG (JTC    1/SC 29/WG 11).-   Non-Patent Literature 2: “Algorithm description for Versatile Video    Coding and Test Model 1 (VTM 1)”; published by Joint Video Experts    Team (JVET) of ITU-T SG 16 WP 3 and ISO/IEC JTC 1/SC 29/WG 11, 10th    Meeting: San Diego, US, Apr. 10-20, 2018 (Document: JVET-J1002-v2).

Problem(s) to be Solved

However, a need exists for an 8-bit core in the VVC to reducecomplexity, reduce memory for storing transform cores and/or reduceinput bit-width of multiplier/add/sub operations. In general, the 2-,4-, 8-, 16- and 32-point DCT-2 transform core, which are 8-bit, cannotbe extracted from, for example, a 64-point DCT-2 transform core, whichis 10-bit in conventional VVC, thus inevitably increasing the cost ofapplying primary transform because a separate 64-point transform corewould need extra storage and logic for implementation. Comparing to10-bit primary transform cores, the 8-bit primary transform core reducesthe complexity because of the reduced memory for storing transform coresand reduced input bit-width of multiplier/add/sub operations.

SUMMARY

According to an aspect of the disclosure, a method of utilizing an 8-bitprimary transform core matrix to decode compressed video or image dataor encode uncompressed video or image data may include: receivinginformation regarding a target data block for encoding or decoding, thetarget data block for encoding or decoding being one of: a compressedvideo or image data block or an uncompressed video or image data block,and determining, for the encoding or decoding of the target data block,whether to use a first transform core matrix that is a DST-7 primarytransform core matrix which is larger than or equal to 8-point andrepresented by 8-bits. The method according to the aspect of thedisclosure may further include: when a result of the determination is touse the first transform core matrix: (i) deriving the 8-bit DST-7primary transform core matrix based on a set of M unique numbers, and(ii) causing or transmitting information that causes the target datablock for encoding or decoding to be encoded or decoded using thederived 8-bit DST-7 primary transform core matrix, and when a result ofthe determination is to not use the first transform core matrix: (i)deriving a second primary transform core matrix based on the same set ofM unique numbers, and (ii) causing or transmitting information thatcauses the target data block for encoding or decoding to be encoded ordecoded using the derived second primary transform core matrix, whereinthe second primary transform core matrix is one of DCT-8, DST-1 andDCT-5.

According to an aspect of the disclosure, an apparatus for utilizing an8-bit primary transform core matrix to decode compressed video or imagedata or encode uncompressed video or image data may include: at leastone memory configured to store computer program code, and at least oneprocessor configured to access the at least one memory and operateaccording to the computer program code. The computer program code maycomprise first applying code, second applying code, third applying code,fourth applying code, fifth applying code and sixth applying code, butthe disclosure is not so limited and may include more sets of applyingcode.

The first applying code may be configured to cause the at least oneprocessor to receive information regarding a target data block forencoding or decoding, the target data block for encoding or decodingbeing one of: a compressed video or image data block or an uncompressedvideo or image data block. The second applying code may be configured tocause the at least one processor to determine, for the encoding ordecoding of the target data block, whether to use a first transform corematrix that is a DST-7 primary transform core matrix that is larger thanor equal to 8-point and represented by 8-bits.

The third applying code may be configured cause the at least oneprocessor to when a result of the determination is to use the firsttransform core matrix: (i) derive the 8-bit DST-7 primary transform corematrix based on a set of M unique numbers, and (ii) cause or transmitinformation that causes the target data block for encoding or decodingto be encoded or decoded using the derived 8-bit DST-7 primary transformcore matrix. The fourth applying code configured to cause the at leastone processor to when a result of the determination is to not use thefirst primary transform core matrix: (i) derive a second primarytransform core matrix based on the same set of M unique numbers, and(ii) cause or transmit information that causes the target data block forencoding or decoding to be encoded or decoded using the derived secondprimary transform core matrix, wherein the second primary transform corematrix is one of DCT-8, DST-1 and DCT-5.

According to an aspect of the disclosure, a non-transitorycomputer-readable storage medium storing may store instructions thatcause one or more processors to: receive information regarding a targetdata block for encoding or decoding, the target data block for encodingor decoding being one of: a compressed video or image data block or anuncompressed video or image data block, and determine, for the encodingor decoding of the target data block, whether to use a first transformcore matrix that is a DST-7 primary transform core matrix that is largerthan or equal to 8-point and represented by 8-bits.

The stored instructions of the non-transitory computer-readable storagemedium may further cause the one or more processors to: when a result ofthe determination is to use the first transform core matrix: (i) derivethe 8-bit DST-7 primary transform core matrix based on a set of M uniquenumbers, and (ii) cause or transmit information that causes the targetdata block for encoding or decoding to be encoded or decoded using thederived 8-bit DST-7 primary transform core matrix, and when a result ofthe determination is to not use the first primary transform core matrix:(i) derive a second primary transform core matrix based on the same setof M unique numbers, and (ii) cause or transmit information that causesthe target data block for encoding or decoding to be encoded or decodedusing the derived second primary transform core matrix, wherein thesecond primary transform core matrix is one of DCT-8, DST-1 and DCT-5.

BRIEF DESCRIPTION OF THE DRAWINGS

Further features, the nature, and various advantages of the disclosedsubject matter will be more apparent from the following detaileddescription and the accompanying drawings in which:

FIG. 1 is a schematic illustration of a simplified block diagram of acommunication system in accordance with an embodiment.

FIG. 2 is a schematic illustration of a simplified block diagram of astreaming system in accordance with an embodiment.

FIG. 3 is a schematic illustration of a simplified block diagram of adecoder in accordance with an embodiment.

FIG. 4 is a schematic illustration of a simplified block diagram of anencoding system including an encoder and a local decoder in accordancewith an embodiment.

FIG. 5 is a schematic illustration of an exemplary method of encoding ordecoding in accordance with an embodiment.

FIG. 6 is a schematic illustration of outputting a transform core matrixin accordance with an embodiment.

FIG. 7 is a schematic illustration of a computer system in accordancewith an embodiment.

DETAILED DESCRIPTION

FIG. 1 illustrates a simplified block diagram of a communication system(100) according to an embodiment of the present disclosure. The system(100) may include at least two terminals (110, 120) interconnected via anetwork (150). For unidirectional transmission of data, a first terminal(110) may code video data at a local location for transmission to theother terminal (120) via the network (150). The second terminal (120)may receive the coded video data of the other terminal from the network(150), decode the coded data and display the recovered video data.Unidirectional data transmission may be common in media servingapplications and the like.

FIG. 1 illustrates a second pair of terminals (130, 140) provided tosupport bidirectional transmission of coded video that may occur, forexample, during videoconferencing. For bidirectional transmission ofdata, each terminal (130, 140) may code video data captured at a locallocation for transmission to the other terminal via the network (150).Each terminal (130, 140) also may receive the coded video datatransmitted by the other terminal, may decode the coded data and maydisplay the recovered video data at a local display device.

In FIG. 1 , the terminals (110-140) may be illustrated as a laptop 110,a server 120, and smartphones 130 and 140, but the principles of thepresent disclosure are not so limited. Embodiments of the presentdisclosure find application with other equipment including but notlimited to laptop computers, tablet computers, media players and/ordedicated video conferencing equipment. The network (150) represents anynumber of networks that convey coded video data among the terminals(110-140), including for example wireline and/or wireless communicationnetworks. The communication network (150) may exchange data incircuit-switched and/or packet-switched channels. Representativenetworks include telecommunications networks, local area networks, widearea networks and/or the Internet. For the purposes of the presentdiscussion, the architecture and topology of the network (150) may beimmaterial to the operation of the present disclosure unless explainedherein below.

FIG. 2 illustrates, as an example for an application for the disclosedsubject matter, the placement of a decoder 433 in a streamingenvironment/streaming system 200. The disclosed subject matter can beequally applicable to other video enabled applications, including, forexample, video conferencing, digital TV, storing of compressed video ondigital media including CD, DVD, memory stick and the like, and so on.

As shown in FIG. 2 , a streaming system (200) may include a capturesubsystem (213), which can include a video source (201), for example adigital camera, which may create an uncompressed video sample stream(202). That sample stream (202), depicted as a bold line to emphasize ahigh data volume when compared to encoded video bit streams, can beprocessed by a source encoder (203) coupled to the camera (201). Thesource encoder (203) can include hardware (e.g., a processor or circuitand memory), software, or a combination thereof to enable or implementaspects of the disclosed subject matter as described in more detailbelow. The encoded video bit stream (204), depicted as a thin line toemphasize the lower data volume when compared to the sample stream, canbe stored on a streaming server (205) for future use. One or morestreaming clients (206, 208) can access the streaming server (205) toretrieve copies (207, 209) of the encoded video bitstream (204). Aclient (206) can include a video decoder (210) which may decode theincoming copy of the encoded video bitstream (207) and may create anoutgoing video sample stream (211) that can be rendered on a display(212) or other rendering device (not depicted).

FIG. 3 illustrates a functional block diagram of a decoder (210) (e.g.,a video decoder) according to an embodiment of the present disclosure.As shown in FIG. 3 , a receiver (310) may receive one or more codecvideo sequences to be decoded by the decoder (210). In the same oranother embodiment, one coded video sequence may be received at a time,where the decoding of each coded video sequence is independent fromother coded video sequences. The coded video sequence may be receivedfrom a channel (312), which may be a hardware/software link to a storagedevice which stores the encoded video data. The receiver (310) mayreceive the encoded video data with other data, for example, coded audiodata and/or ancillary data streams, that may be forwarded to theirrespective using entities (not depicted). The receiver (310) mayseparate the coded video sequence from the other data. To combat networkjitter, a buffer memory (315) may be coupled in between a receiver (310)and an entropy decoder/parser (320) (“parser” henceforth). When receiver(310) is receiving data from a store/forward device of sufficientbandwidth and controllability, or from an isosynchronous network, thebuffer (315) may not be needed, or can be small. For use on best effortpacket networks such as the Internet, the buffer (315) may be required,which can be comparatively large and can advantageously be of adaptivesize.

The video decoder (210) may include a parser (320) to reconstructsymbols (321) from the entropy coded video sequence. Categories of thosesymbols include information used to manage operation of the decoder(210), and potentially information to control a rendering device such asa display (212) that is not an integral part of the decoder but can becoupled to it, as was shown in FIGS. 2 and 3 . The control informationfor the rendering device(s) may be in the form of SupplementaryEnhancement Information (SEI messages) or Video Usability Information(VUI) parameter set fragments (not depicted). The parser (320) mayparse/entropy-decode the coded video sequence received. The coding ofthe coded video sequence can be in accordance with a video codingtechnology or standard, and can follow principles well known to a personskilled in the art, including variable length coding, Huffman coding,arithmetic coding with or without context sensitivity, and so forth. Theparser (320) may extract from the coded video sequence, a set ofsubgroup parameters for at least one of the subgroups of pixels in thevideo decoder, based upon at least one parameters corresponding to thegroup. Subgroups can include Groups of Pictures (GOPs), pictures, tiles,slices, macroblocks, Coding Units (CUs), blocks, Transform Units (TUs),Prediction Units (PUs) and so forth. The entropy decoder/parser 320 mayalso extract from the coded video sequence information such as transformcoefficients, quantizer parameter values, motion vectors, and so forth.

The parser (320) may perform entropy decoding/parsing operation on thevideo sequence received from the buffer (315), so as to create symbols(321). Reconstruction of the symbols (321) can involve multipledifferent units depending on the type of the coded video picture orparts thereof (such as: inter and intra picture, inter and intra block),and other factors. Which units are involved, and how, can be controlledby the subgroup control information that was parsed from the coded videosequence by the parser (320). The flow of such subgroup controlinformation between the parser (320) and the multiple units below is notdepicted for clarity.

Beyond the functional blocks already mentioned, the decoder 210 can beconceptually subdivided into a number of functional units as describedbelow. In a practical implementation operating under commercialconstraints, many of these units interact closely with each other andcan, at least partly, be integrated into each other. However, for thepurpose of describing the disclosed subject matter, the conceptualsubdivision into the functional units below is appropriate.

A first unit may be the scaler/inverse transform unit (351). Thescaler/inverse transform unit (351) may receive a quantized transformcoefficient as well as control information, including which transform touse, a block size, a quantization factor, quantization scaling matrices,etc. as symbol(s) (321) from the parser (320). It can output blockscomprising sample values that can be input into an aggregator (355).

In some cases, the output samples of the scaler/inverse transform (351)can pertain to an intra-coded block, which is a block that is not usingpredictive information from previously reconstructed pictures, but canuse predictive information from previously reconstructed parts of thecurrent picture. Such predictive information can be provided by an intrapicture prediction unit (352). In some cases, the intra pictureprediction unit (352) generates a block of the same size and shape ofthe block under reconstruction, using surrounding already reconstructedinformation fetched from the current (partly reconstructed) picture(358). The aggregator (355), in some cases, adds, on a per sample basis,the prediction information the intra prediction unit (352) has generatedto the output sample information as provided by the scaler/inversetransform unit (351).

In other cases, the output samples of the scaler/inverse transform unit(351) can pertain to an inter-coded block, and a potentially motioncompensated block. In such a case, a Motion Compensation Prediction unit(353) can access reference a picture memory (357) to fetch samples usedfor prediction. After motion compensating the fetched samples inaccordance with the symbols (321) pertaining to the block, these samplescan be added by the aggregator (355) to the output of the scaler/inversetransform unit (in this case called the residual samples or residualsignal) so to generate output sample information. The addresses withinthe reference picture memory form where the motion compensation unitfetches prediction samples can be controlled by motion vectors,available to the motion compensation unit in the form of symbols (321)that can have, for example X, Y, and reference picture components.Motion compensation also can include interpolation of sample values asfetched from the reference picture memory when sub-sample exact motionvectors are in use, motion vector prediction mechanisms, and so forth.

The output samples of the aggregator (355) can be subject to variousloop filtering techniques in the loop filter unit (356). Videocompression technologies can include in-loop filter technologies thatare controlled by parameters included in the coded video bitstream andmade available to the loop filter unit (356) as symbols (321) from theparser (320), but can also be responsive to meta-information obtainedduring the decoding of previous (in decoding order) parts of the codedpicture or coded video sequence, as well as responsive to previouslyreconstructed and loop-filtered sample values.

The output of the loop filter unit (356) can be a sample stream that canbe output to the render device (212) as well as stored in the referencepicture memory (357) for use in future inter-picture prediction.

Certain coded pictures, once fully reconstructed, can be used asreference pictures for future prediction. Once a coded picture is fullyreconstructed and the coded picture has been identified as a referencepicture (by, for example, parser (320)), the current reference picture(358) can become part of the reference picture buffer (357), and a freshcurrent picture memory can be reallocated before commencing thereconstruction of the following coded picture.

The video decoder 210 may perform decoding operations according to apredetermined video compression technology that may be documented in astandard, such as ITU-T Rec. H.265. The coded video sequence may conformto a syntax specified by the video compression technology or standardbeing used, in the sense that it adheres to the syntax of the videocompression technology or standard, as specified in the videocompression technology document or standard and specifically in theprofiles document therein. Also necessary for compliance can be that thecomplexity of the coded video sequence is within bounds as defined bythe level of the video compression technology or standard. In somecases, levels restrict the maximum picture size, maximum frame rate,maximum reconstruction sample rate (measured in, for example megasamplesper second), and/or maximum reference picture size. Limits set by levelscan, in some cases, be further restricted through Hypothetical ReferenceDecoder (HRD) specifications and metadata for HRD buffer managementsignaled in the coded video sequence.

In an embodiment, the receiver (310) may receive additional (redundant)data with the encoded video. The additional data may be included as partof the coded video sequence(s). The additional data may be used by thevideo decoder (210) to properly decode the data and/or to moreaccurately reconstruct the original video data. Additional data can bein the form of, for example, temporal, spatial, or SNR enhancementlayers, redundant slices, redundant pictures, forward error correctioncodes, and so on.

FIG. 4 may be a functional block diagram of an encoding system (400)including a source coder (203), which may be a video encoder (203)according to an embodiment of the present disclosure.

The encoder (203) may receive video samples from a video source (201)(that is not a part of the encoder) that may capture video image(s) tobe coded by the encoder (203).

The video source (201) may provide the source video sequence to be codedby the encoder (203) in the form of a digital video sample stream thatcan be of any suitable bit depth (for example: 8 bit, 10 bit, 12 bit, .. . ), any colorspace (for example, BT.601 Y CrCB, RGB, . . . ) and anysuitable sampling structure (for example Y CrCb 4:2:0, Y CrCb 4:4:4). Ina media serving system, the video source (201) may be a storage device,which may include a memory and processor, that stores previouslyprepared video. In a videoconferencing system, the video source (201)may include a camera that captures local image information as a videosequence. Video data may be provided as a plurality of individualpictures that impart motion when viewed in sequence. The picturesthemselves may be organized as a spatial array of pixels, wherein eachpixel can comprise one or more sample depending on the samplingstructure, color space, etc. in use. A person skilled in the art canreadily understand the relationship between pixels and samples. Thedescription below focusses on samples.

According to an embodiment, the encoder (203) may code and compress thepictures of the source video sequence into a coded video sequence inreal time or under any other time constraints as required by theapplication. Enforcing appropriate coding speed is one function ofcontroller (450). The controller 450 may control other functional unitsas described below and is functionally coupled to these units. Thecoupling is not depicted for clarity. Parameters set by the controller450 can include rate control related parameters (e.g., picture skip,quantizer, lambda value of rate-distortion optimization techniques), apicture size, a group of pictures (GOP) layout, a maximum motion vectorsearch range, and so forth. A person skilled in the art can readilyidentify other functions of controller (450) as they may pertain tovideo encoder (203) optimized for a certain system design.

Some video encoders operate in what a person skilled in the art readilyrecognizes as a “coding loop.” As an oversimplified description, acoding loop can consist of the encoding part of a source encoder (430)(“source coder” or “source encoder” henceforth), which is responsiblefor creating symbols based on an input picture to be coded, and areference picture(s), and a local decoder (433) embedded in the encoder(203) that reconstructs the symbols to create the sample data a remotedecoder 210 also would create (as any compression between symbols andcoded video bitstream is lossless in the video compression technologiesconsidered in the disclosed subject matter). That reconstructed samplestream is input to the reference picture memory (434). As the decodingof a symbol stream leads to bit-exact results independent of a decoderlocation (local or remote), the reference picture buffer content is alsobit exact between local encoder and remote encoder. In other words, theprediction part of an encoder “sees,” as reference picture samples,exactly the same sample values as a decoder would “see” when usingprediction during decoding. This fundamental principle of referencepicture synchronicity (and resulting drift, if synchronicity cannot bemaintained, for example, because of channel errors) is well known to aperson skilled in the art.

The operation of the “local” decoder (433) can be the same as of a“remote” decoder (210), which has already been described in detail abovein conjunction with FIG. 3 . Briefly referring also to FIG. 3 , however,as symbols are available and en/decoding of symbols to a coded videosequence by entropy coder (445) and parser (320) can be lossless, theentropy decoding parts of decoder (210) (including channel (312),receiver (310), buffer (315), and parser (320)) may not be fullyimplemented in local decoder (433).

An observation that can be made at this point is that any decodertechnology except the parsing/entropy decoding that is present in adecoder also necessarily needs to be present, in substantially identicalfunctional form, in a corresponding encoder. The description of encodertechnologies are the inverse of the described decoder technologies. Onlyin certain areas a more detail description is required and providedbelow.

As part of its operation, the source coder (430) may perform motioncompensated predictive coding, which codes an input frame predictivelywith reference to one or more previously-coded frames from the videosequence that were designated as “reference frames.” In this manner, thecoding engine (432) codes differences between pixel blocks of an inputframe and pixel blocks of reference frame(s) that may be selected asprediction reference(s) to the input frame.

The local video decoder (433) may decode coded video data of frames thatmay be designated as reference frames, based on symbols created by thesource coder (430). Operations of the coding engine (432) mayadvantageously be lossy processes. When the coded video data may bedecoded at a video decoder (not shown in FIG. 4 ), the reconstructedvideo sequence typically may be a replica of the source video sequencewith some errors. The local video decoder (433) replicates decodingprocesses that may be performed by the video decoder on reference framesand may cause reconstructed reference frames to be stored in thereference picture cache (434). In this manner, the source encoder (203)may store copies of reconstructed reference frames locally that havecommon content as the reconstructed reference frames that will beobtained by a far-end video decoder (absent transmission errors).

The predictor (435) may perform prediction searches for the codingengine (432). That is, for a new frame to be coded, the predictor (435)may search the reference picture memory (434) for sample data (ascandidate reference pixel blocks) or certain metadata such as referencepicture motion vectors, block shapes, and so on, that may serve as anappropriate prediction reference for the new pictures. The predictor(435) may operate on a sample block-by-pixel block basis to findappropriate prediction references. In some cases, as determined bysearch results obtained by the predictor (435), an input picture mayhave prediction references drawn from multiple reference pictures storedin the reference picture memory (434).

The controller (450) may manage coding operations of the source coder(203), including, for example, setting of parameters and subgroupparameters used for encoding the video data.

Output of all aforementioned functional units may be subjected toentropy coding in the entropy coder (445). The entropy coder maytranslate the symbols as generated by the various functional units intoa coded video sequence, by loss-less compressing the symbols accordingto technologies known to a person skilled in the art as, for exampleHuffman coding, variable length coding, arithmetic coding, and so forth.

The transmitter (440) may buffer the coded video sequence(s) as createdby the entropy coder (445) to prepare it for transmission via acommunication channel (460), which may be a hardware/software link to astorage device which would store the encoded video data. The transmitter(440) may merge coded video data from the source coder (203) with otherdata to be transmitted, for example, coded audio data and/or ancillarydata streams (sources not shown).

The controller (450) may manage operation of the encoder (203). Duringcoding, the controller (450) may assign to each coded picture a certaincoded picture type, which may affect the coding techniques that may beapplied to the respective picture. For example, pictures often may beassigned as one of the following frame types from among, for example, anIntra Picture (I picture), a Predictive picture (P picture) and aBi-directionally Predictive Picture (B Picture).

An Intra Picture (I picture) may be one that may be coded and decodedwithout using any other frame in the sequence as a source of prediction.Some video codecs allow for different types of Intra pictures,including, for example Independent Decoder Refresh Pictures. A personskilled in the art is aware of those variants of I pictures and theirrespective applications and features.

A Predictive Picture (P picture) may be one that may be coded anddecoded using intra prediction or inter prediction using at most onemotion vector and reference index to predict the sample values of eachblock.

A Bi-directionally Predictive Picture (B Picture) may be one that may becoded and decoded using intra prediction or inter prediction using atmost two motion vectors and reference indices to predict the samplevalues of each block. Similarly, multiple-predictive pictures can usemore than two reference pictures and associated metadata for thereconstruction of a single block.

Source pictures commonly may be subdivided spatially into a plurality ofsample blocks (for example, blocks of 4×4, 8×8, 4×8, or 16×16 sampleseach) and coded on a block-by-block basis. Blocks may be codedpredictively with reference to other (already coded) blocks asdetermined by the coding assignment applied to the blocks' respectivepictures. For example, blocks of I pictures may be codednon-predictively or they may be coded predictively with reference toalready coded blocks of the same picture (spatial prediction or intraprediction). Pixel blocks of P pictures may be coded predictively, viaspatial prediction or via temporal prediction with reference to onepreviously coded reference pictures. Blocks of B pictures may be codedpredictively, via spatial prediction or via temporal prediction withreference to one or two previously coded reference pictures.

The source coder (203) may perform coding operations according to apredetermined video coding technology or standard, such as ITU-T Rec.H.265. In its operation, the source coder (203) may perform variouscompression operations, including predictive coding operations thatexploit temporal and spatial redundancies in the input video sequence.The coded video data, therefore, may conform to a syntax specified bythe video coding technology or standard being used.

In an embodiment, the transmitter (440) may transmit additional datawith the encoded video. The source coder (430) may include such data aspart of the coded video sequence. Additional data may comprisetemporal/spatial/SNR enhancement layers, other forms of redundant datasuch as redundant pictures and slices, Supplementary EnhancementInformation (SEI) messages, Visual Usability Information (VUI) parameterset fragments, and so on.

As discussed above, the coding of the coded video sequence can be inaccordance with a video coding technology or standard, and can followprinciples well known to a person skilled in the art, including variablelength coding, Huffman coding, arithmetic coding with or without contextsensitivity, and so forth. In some streaming systems, the videobitstreams (204, 207, 209) can be encoded according to certain videocoding/compression standards. An example of those standards includeITU-T HEVC Recommendation of H.265.

In HEVC, the primary transforms may have 4-point, 8-point, 16-point or32-point DCT-2 transform core matrices, and the transform core matricesmay be represented using 8-bit integers, (i.e., an 8-bit transform corematrix). The HEVC transform core matrices of smaller DCT-2 are part oflarger DCT-2, as shown below:

4×4 DCT-2 Transform

-   -   {64, 64, 64, 64}    -   {83, 36, −36, −83}    -   {64, −64, −64, 64}    -   {36, −83, 83, −36}

8×8 DCT-2 Transform

-   -   {64, 64, 64, 64, 64, 64, 64, 64}    -   {89, 75, 50, 18, −18, −50, −75, −89}    -   {83, 36, −36, −83, −83, −36, 36, 83}    -   {75, −18, −89, −50, 89, 18, −75}    -   {64, −64, −64, 64, 64, −64, −64, 64}    -   {50, −89, 18, 75, −75, −18, 89, −50}    -   {36, −83, 83, −36, −36, 83, 83, −36}    -   {18, −50, 75, −89, 89, −75, 50, −18}

16×16 DCT-2 Transform

-   -   {64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64}    -   {90 87 80 70 57 43 25 9 −9−25−43−57−70−80−87−90}    -   {89 75 50 18−18−50−75−89−89−75−50−18 18 50 75 89}    -   {87 57 9−43−80−90−70−25 25 70 90 50 43−9−57−87}    -   {83 36−36−83−83−36 36 83 83 36−36−83−83−36 36 83}    -   {80 9−70−87−25 57 90 43−43−90−57 25 67 70−9−80}    -   {75−18−89−50 50 89, 18−75−75 18 89 50−50−89−18 75}    -   {70−43−87 9 90 25−8−57 57 80−25−90−9 87 43−70}    -   {64−64−64 64 64−64−64 64 64−64−64 64 64−64−64 64}    -   {57−80−25 90−9−87 43 70−70−43 87 9−90 25 80−57}    -   {50−89 18 75−75−18 89−50−50, 89−18−75 75 18−89 50}    -   {43−90 57 25−87 70 9−80 80−9−70 87−25−57 90−43}    -   {36−83 83−36−36 83−87 36 36−83 83−36−36 83−83 36}    -   {25−70 90−80 43 9−57 87−87 57−9−43 80−90 70−25}    -   {18−50 75−89 89−75 50−18−18 50−75 89−89 75−50 18}    -   {9−25 43−57 70−80 87−90 90−87 80−70 57−43 25−9}

32×32 DCT-2 Transform

-   -   {64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64        64 64 64 64 64 64 64 64 64 64 64}    -   {90 90 88 85 82 78 73 67 61 54 46 38 31 22 13 4        −4−13−22−31−38−46−54−61−67−73−78−82−85−88−90−90}    -   {90 87 80 70 57 43 25 9        −9−25−43−57−70−80−87−90−90−87−80−70−57−43−25−9 9 25 43 57 70 80        87 90}    -   {90 82 67 46 22 −4−31−54−73−85−90−88−78−61−38−13 13 38 61 78 88        90 85 73 54 31 4−22−46−67−82−90}    -   {89 75 50 18 −18−50−75−89−89−75−50−18 18 50 75 89 89 75 50        18−18−50−75−89−89−75−50−18 18 50 75 89}    -   {88 67 31−13−54−82−90−78−46−4 38 73 90 85 61 22 22 61 85 90 73        38 4 46 78 90 82 54 13−31−67−88}    -   {87 57 9−43−80−90−70−25 25 70 90 80 43−9−57−87−87−57−9 43 80 90        70 25−25−70−90−80−43 9 57 87}    -   {85 46 13 67 90 73 22 38 82 88 54−4−61−90−76−31 31 78 90 61        4−54−88−82−38 22 73 90 67 13−46−85}    -   {83 36−36−83−83−36 36 83 83 36−36−83−83−36 36 83 83        36−36−83−83−36 36 83 83 36−36−83−83−36 36 83}    -   {82 22−54−90−61 13 78 85 31−46−90−67 4 73 88 38−38−88−73−4 67 90        46−31−85−78−13 61 90 54−22−82}    -   {80 9−70−87−25 57 90 43−43−90−57 25 87 70−9−80−60−9 70 87        25−57−90−43 43 90 57−25−87−70 9 80}    -   {76−4−82−73 13 85 67−22−88−61 31 90 54−38−90−46 46 90        38−54−90−31 61 88 22−67−85−13 73 82 4−78}    -   {75−18−89−50 50 89 18−75−75 18 89 50−50−89−18 75 75−18−89−50 50        89 18−75−75 18 89 50−50−89−16 75}    -   {73−31−90−22 78 67−38−90−13 82 61−46−88−4 85 54−54−85 4 88        46−61−82 13 90 38−67−78 22 90 31−73}    -   {70−43−87 9 90 25−80−57 57 80−25−90−9 87 43−70−70 43 87−9−90−25        80 57−57−80 25 90 9−87−43 70}    -   {67−54−76 38 85−22−90 4 90 13−88−31 82 46−73−61 61 73−46−82 31        88−13−90−4 90 22−85−38 78 54−67}    -   {64−64−64 64 64−64−64 64 64−64−64 64 64−64−64 64 64−64−64 64        64−64−64 64 64−64−64 64 64−64−64 64}    -   {61−73−46 82 31−88−13 90−4−90 22 85−38−78 54 67−67−54 78        38−85−22 90 4−90 13 88−31−82 46 73−61}    -   {57−80−25 90−9−87 43 70−70−43 87 9−90 25 80−57−57 80 25−90 9        67−43−70 70 43−87−9 90−25−80 57}    -   {54−05 −4 68−46−61 82 13−90 38 67−78−22 90−31−73 73 31−90 22        78−67−38 90−13−82 61 46−88 4 85−54}    -   {50−89 18 75−75−18 89−50−50 89−18−75 75 18−89 50 50−89 18        75−75−18 89−50−50 89−18−75 75 18−89 50}    -   {46−90 38 54−90 31 61−68 22 67−85 13 73−82 4 78−78−4 82−73−13        85−67−22 88−61−31 90−54−38 90−46}    -   {43−90 57 25−87 70 9−80 80−9−70 87−25−57 90−43−43 90−57−25        87−70−9 80−80 9 70−87 25 57−90 43}    -   {38−88 73−4−67 90−46−31 85−78 13 61−90 54 22−82 82−22−54        90−61−13 78−85 31 46−90 67 4−73 88−38}    -   {36−83 63−36−36 83−63 36 36−83 83−36−36 83−83 36 36−83 83−36−36        83−63 36 36−83 83−36−36 83−83 36}    -   {31−78 90−61 4 54−88 82−36−22 73−90 67−13−46 85−85 46 13−67        90−73 22 38−82 88−54−4 61−90 78−31}    -   {25−70 90−80 43 9−57 87−07 57−9−43 80−90 70−25−25 70−90 80−43−9        57−87 87−57 9 43−80 90−70 25}    -   {22−61 85−90 73−38−4 46−78 90−82 54−13−31 67−88 88−67 31 13−54        82−90 76−46 4 38−73 90−85 61−22}    -   {18−50 75−89 69−75 50−18−18 50−75 89−89 75−50 18 18−50 75−89        89−75 50−18−18 50−75 89−89 75−50 18}    -   {13−38 61−76 88−90 85−73 54−31 4 22−46 67−82 90−90 82−67 46−22−4        31−54 73−85 90−88 78−61 38−13}    -   {9−25 43−57 70−80 87−90 90−87 80−70 57−43 25−9 −9 25−43 57−70        80−87 90−90 87−80 70−57 43−25 9}    -   {4−13 22−31 38−46 54−61 67−73 78−82 85−88 90−90 90−90 88−85        82−78 73−67 61−54 46−38 31−22 13−4}

As shown above, each of the DCT-2 cores show symmetry/anti-symmetrycharacteristics. Thus, a so-called “partial butterfly” implementationmay be supported to reduce the number of operation counts(multiplications, adds/subs, shifts), and identical results of matrixmultiplication can be obtained using a partial butterfly implementation.More details regarding HEVC transform design can be found in Non-PatentLiterature 1.

Under development is a video coding standard informally known asVersatile Video Coding (VVC). The disclosed subject matter may be usedin the context of VVC.

For a primary transform in VVC, besides 4-point, 8-point, 16-point and32-point DCT-2 transforms, which are the same as with HEVC, additional2-point, 64-point and 128-point DCT-2 may also be included. The 2-pointDCT-2 core can be an 8-bit integer matrix which is aligned with the HEVCprimary transform design, but the 64-point and 128-point DCT-2 use10-bit transform cores, which means the 2-point, 4-point, 8-point,16-point and 32-point DCT-2 transforms cannot be extracted from the64-point DCT-2 core or 128-point DCT-2 core in VVC.

Parts of an exemplary 64-point DCT-2 core defined in VVC are shownbelow, where

-   -   {aa, ab, ac, ad, ae, af, ag, ah, ai, aj, ak, al, am, an, ao, ap,        aq, ar, as, at, au, av, aw, ax, ay, az, ba, bb, bc, bd, be, bf,        bg, bh, bi, bj, bk, bl, bm, bn, bo, bp, bq, br, bs, bt, bu, by,        bw, bx, by, bz, ca, cb, cc, cd, ce, cf, cg, ch, ci, cj,        ck}={256, 334, 139, 355, 301, 201, 71, 360, 346, 319, 280, 230,        171, 105, 35, 362, 358, 351, 341, 327, 311, 291, 268, 243, 216,        186, 155, 122, 88, 53, 18, 362, 361, 359, 357, 353, 349, 344,        338, 331, 323, 315, 306, 296, 285, 274, 262, 250, 236, 223, 208,        194, 178, 163, 147, 130, 114, 97, 79, 62, 44, 27, 9}:    -   {aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa,        aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa,        aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa,        aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa}    -   {bf, bg, bh, bi, bj, bk, bl, bm, bn, bo, bp, bq, br, bs, bt, bu,        by, bw, bx, by, bz, ca, cb, cc, cd, ce, cf, cg, ch, ci, cj, ck,        −ck, −cj, −ci, −ch, −cg, −cf, −ce, −cd, −cc, −cb, −ca, −bz, −by,        −bx, −bw, −by, −bu, −bt, −bs, −br, −bq, −bp, −bo, −bn, −bm, −bl,        −bk, −bj, −bi, −bh, −bg, −bf}    -   {ap, aq, ar, as, at, au, av, aw, ax, ay, az, ba, bb, bc, bd, be,        −be, −bd, −bc, −bb, −ba, −az, −ay, −ax, −aw, −av, −au, −at, −as,        −ar, −aq, −ap, −ap, −aq, −ar, −as, −at, −au, −av, −aw, −ax, −ay,        −az, −ba, −bb, −bc, −bd, −be, be, bd, bc, bb, ba, az, ay, ax,        aw, av, au, at, as, ar, aq, ap}    -   {bg, bj, bm, bp, bs, by, by, cb, ce, ch, ck, −ci, −cf, −cc, −bz,        −bw, −bt, −bq, −bn, −bk, −bh, −bf, −bi, −bl, −bo, −br, −bu, −bx,        −ca, −cd, −cg, −cj, cj, cg, cd, ca, bx, bu, br, bo, bl, bi, bf,        bh, bk, bn, bq, bt, bw, bz, cc, cf, ci, −ck, −ch, −ce, −cb, −by,        −by, −bs, −bp, −bm, −bj, −bg}    -   {ah, ai, aj, ak, al, am, an, ao, −ao, −an, −am, −al, −ak, −aj,        −ai, −ah, −ah, −ai, −aj, −ak, −al, −am, −an, −ao, ao, an, am,        al, ak, aj, ai, ah, ah, ai, aj, ak, al, am, an, ao, −ao, −an,        −am, −al, −ak, −aj, −ai, −ah, −ah, −ai, −aj, −ak, −al, −am, −an,        −ao, ao, an, am, al, ak, aj, ai, ah}    -   {bh, bm, br, bw, cb, cg, −ck, −cf, −ca, −by, −bq, −bl, −bg, −bi,        −bn, −bs, −bx, −cc, −ch, cj, ce, bz, bu, bp, bk, bf, bj, bo, bt,        by, cd, ci, −ci, −cd, −by, −bt, −bo, −bj, −bf, −bk, −bp, −bu,        −bz, −ce, −cj, ch, cc, bx, bs, bn, bi, bg, bl, bq, by, ca, cf,        ck, −cg, −cb, −bw, −br, −bm, −bh}    -   {aq, at, aw, az, bc, −be, −bb, −ay, −av, −as, −ap, −ar, −au,        −ax, −ba, −bd, bd, ba, ax, au, ar, ap, as, av, ay, bb, be, −bc,        −az, −aw, −at, −aq, −aq, −at, −aw, −az, −bc, be, bb, ay, av, as,        ap, ar, au, ax, ba, bd, −bd, −ba, −ax, −au, −ar, −ap, −as, −av,        −ay, −bb, −be, bc, az, aw, at, aq}    -   {bi, bp, bw, cd, ck, −ce, −bx, −bq, −bj, −bh, −bo, −by, −cc,        −cj, cf, by, br, bk, bg, bn, bu, cb, ci, −cg, −bz, −bs, −bl,        −bf, −bm, −bt, −ca, −ch, ch, ca, bt, bm, bf, bl, bs, bz, cg,        −ci, −cb, −bu, −bn, −bg, −bk, −br, −by, −cf, cj, cc, by, bo, bh,        bj, bq, bx, ce, −ck, −cd, −bw, −bp, −bi}    -   {ad, ae, af, ag, −ag, −af, −ae, −ad, −ad, −ae, −af, −ag, ag, af,        ae, ad, ad, ae, af, ag, −ag, −af, −ae, −ad, −ad, −ae, −af, −ag,        ag, af, ae, ad, ad, ae, af, ag, −ag, −af, −ae, −ad, −ad, −ae,        −af, −ag, ag, af, ae, ad, ad, ae, af, ag, −ag, −af, −ae, −ad,        −ad, −ae, −af, −ag, ag, af, ae, ad}    -   {bj, bs, cb, ck, −cc, −bt, −bk, −bi, −br, −ca, −cj, cd, bu, bl,        bh, bq, bz, ci, −ce, −by, −bm, −bg, −bp, −by, −ch, cf, bw, bn,        bf, bo, bx, cg, −cg, −bx, −bo, −bf, −bn, −bw, −cf, ch, by, bp,        bg, bm, by, ce, −ci, −bz, −bq, −bh, −bl, −bu, −cd, cj, ca, br,        bi, bk, bt, cc, −ck, −cb, −bs, −bj}    -   {ar, aw, bb, −bd, −ay, −at, −ap, −au, −az, −be, ba, av, aq, as,        ax, bc, −bc, −ax, −as, −aq, −av, −ba, be, az, au, ap, at, ay,        bd, −bb, −aw, −ar, −ar, −aw, −bb, bd, ay, at, ap, au, az, be,        −ba, −av, −aq, −as, −ax, −bc, bc, ax, as, aq, av, ba, −be, −az,        −au, −ap, −at, −ay, −bd, bb, aw, ar}    -   {bk, by, cg, −ce, −bt, −bi, −bm, −bx, −ci, cc, br, bg, bo, bz,        ck, −ca, −bp, −bf, −bq, −cb, cj, by, bn, bh, bs, cd, −ch, −bw,        −bl, −bj, −bu, −cf, cf, bu, bj, bl, bw, ch, −cd, −bs, −bh, −bn,        −by, −cj, cb, bq, bf, bp, ca, −ck, −bz, −bo, −bg, −br, −cc, ci,        bx, bm, bi, bt, ce, −cg, −by, −bk}    -   {ai, al, ao, −am, −aj, −ah, −ak, −an, an, ak, ah, aj, am, −ao,        −al, −ai, −ai, −al, −ao, am, aj, ah, ak, an, −an, −ak, −ah, −aj,        −am, ao, al, ai, ai, al, ao, −am, −aj, −ah, −ak, −an, an, ak,        ah, aj, am, −ao, −al, −ai, −ai, −al, −ao, am, aj, ah, ak, an,        −an, −ak, −ah, −aj, −am, ao, al, ai}    -   {bl, by, −ck, −bx, −bk, −bm, −bz, cj, bw, bj, bn, ca, −ci, −by,        −bi, −bo, −cb, ch, bu, bh, bp, cc, −cg, −bt, −bg, −bq, −cd, cf,        bs, bf, br, ce, −ce, −br, −bf, −bs, −cf, cd, bq, bg, bt, cg,        −cc, −bp, −bh, −bu, −ch, cb, bo, bi, by, ci, −ca, −bn, −bj, −bw,        −cj, bz, bm, bk, bx, ck, −by, −bl}    -   {as, az, −bd, −aw, −ap, −av, −bc, ba, at, ar, ay, −be, −ax, −aq,        −au, −bb, bb, au, aq, ax, be, −ay, −ar, −at, −ba, bc, av, ap,        aw, bd, −az, −as, −as, −az, bd, aw, ap, av, bc, −ba, −at, −ar,        −ay, be, ax, aq, au, bb, −bb, −au, −aq, −ax, −be, ay, ar, at,        ba, −bc, −av, −ap, −aw, −bd, az, as}    -   {bm, cb, −cf, −bq, −bi, −bx, cj, bu, bf, bt, ci, −by, −bj, −bp,        −ce, cc, bn, bl, ca, −cg, −br, −bh, −bw, ck, by, bg, bs, ch,        −bz, −bk, −bo, −cd, cd, bo, bk, bz, −ch, −bs, −bg, −by, −ck, bw,        bh, br, cg, −ca, −bl, −bn, −cc, ce, bp, bj, by, −ci, −bt, −bf,        −bu, −cj, bx, bi, bq, cf, −cb, −bm}    -   {ab, ac, −ac, −ab, −ab, −ac, ac, ab, ab, ac, −ac, −ab, −ab, −ac,        ac, ab, ab, ac, −ac, −ab, −ab, −ac, ac, ab, ab, ac, −ac, −ab,        −ab, −ac, ac, ab, ab, ac, −ac, −ab, −ab, −ac, ac, ab, ab, ac,        −ac, −ab, −ab, −ac, ac, ab, ab, ac, −ac, −ab, −ab, −ac, ac, ab,        ab, ac, −ac, −ab, −ab, −ac, ac, ab}    -   {bn, ce, −ca, −bj, −br, −ci, bw, bf, by, −cj, −bs, −bi, −bz, cf,        bo, bm, cd, −cb, −bk, −bq, −ch, bx, bg, bu, −ck, −bt, −bh, −by,        cg, bp, bl, cc, −cc, −bl, −bp, −cg, by, bh, bt, ck, −bu, −bg,        −bx, ch, bq, bk, cb, −cd, −bm, −bo, −cf, bz, bi, bs, cj, −by,        −bf, −bw, ci, br, bj, ca, −ce, −bn}    -   {at, bc, −ay, −ap, −ax, bd, au, as, bb, −az, −aq, −aw, be, av,        ar, ba, −ba, −ar, −av, −be, aw, aq, az, −bb, −as, −au, −bd, ax,        ap, ay, −bc, −at, −at, −bc, ay, ap, ax, −bd, −au, −as, −bb, az,        aq, aw, −be, −av, −ar, −ba, ba, ar, av, be, −aw, −aq, −az, bb,        as, au, bd, −ax, −ap, −ay, bc, at}    -   {bo, ch, −by, −bh, −ca, cc, bj, bt, −cj, −bq, −bm, −cf, bx, bf,        by, −ce, −bl, −br, −ck, bs, bk, cd, −bz, −bg, −bw, cg, bn, bp,        ci, −bu, −bi, −cb, cb, bi, bu, −ci, −bp, −bn, −cg, bw, bg, bz,        −cd, −bk, −bs, ck, br, bl, ce, −by, −bf, −bx, cf, bm, bq, cj,        −bt, −bj, −cc, ca, bh, by, −ch, −bo}    -   {aj, ao, −ak, −ai, −an, al, ah, am, −am, −ah, −al, an, ai, ak,        −ao, −aj, −aj, −ao, ak, ai, an, −al, −ah, −am, am, ah, al, −an,        −ai, −ak, ao, aj, aj, ao, −ak, −ai, −an, al, ah, am, −am, −ah,        −al, an, ai, ak, −ao, −aj, −aj, −ao, ak, ai, an, −al, −ah, −am,        am, ah, al, −an, −ai, −ak, ao, aj}    -   {bp, ck, −bq, −bo, −cj, br, bn, ci, −bs, −bm, −ch, bt, bl, cg,        −bu, −bk, −cf, by, bj, ce, −bw, −bi, −cd, bx, bh, cc, −by, −bg,        −cb, bz, bf, ca, −ca, −bf, −bz, cb, bg, by, −cc, −bh, −bx, cd,        bi, bw, −ce, −bj, −by, cf, bk, bu, −cg, −bl, −bt, ch, bm, bs,        −ci, −bn, −br, cj, bo, bq, −ck, −bp}    -   {au, −be, −at, −av, bd, as, aw, −bc, −ar, −ax, bb, aq, ay, −ba,        −ap, −az, az, ap, ba, −ay, −aq, −bb, ax, ar, bc, −aw, −as, −bd,        av, at, be, −au, −au, be, at, av, −bd, −as, −aw, bc, ar, ax,        −bb, −aq, −ay, ba, ap, az, −az, −ap, −ba, ay, aq, bb, −ax, −ar,        −bc, aw, as, bd, −av, −at, −be, au}    -   {bq, −ci, −bl, −by, cd, bg, ca, −by, −bi, −cf, bt, bn, ck, −bo,        −bs, cg, bj, bx, −cb, −bf, −cc, bw, bk, ch, −br, −bp, cj, bm,        bu, −ce, −bh, −bz, bz, bh, ce, −bu, −bm, −cj, bp, br, −ch, −bk,        −bw, cc, bf, cb, −bx, −bj, −cg, bs, bo, −ck, −bn, −bt, cf, bi,        by, −ca, −bg, −cd, by, bl, ci, −bq}    -   {ae, −ag, −ad, −af, af, ad, ag, −ae, −ae, ag, ad, af, −af, −ad,        −ag, ae, ae, −ag, −ad, −af, af, ad, ag, −ae, −ae, ag, ad, af,        −af, −ad, −ag, ae, ae, −ag, −ad, −af, af, ad, ag, −ae, −ae, ag,        ad, af, −af, −ad, −ag, ae, ae, −ag, −ad, −af, af, ad, ag, −ae,        −ae, ag, ad, af, −af, −ad, −ag, ae}    -   {br, −cf, −bg, −cc, bu, bo, −ci, −bj, −bz, bx, bl, ck, −bm, −bw,        ca, bi, ch, −bp, −bt, cd, bf, ce, −bs, −bq, cg, bh, cb, −by,        −bn, cj, bk, by, −by, −bk, −cj, bn, by, −cb, −bh, −cg, bq, bs,        −ce, −bf, −cd, bt, bp, −ch, −bi, −ca, bw, bm, −ck, −bl, −bx, bz,        bj, ci, −bo, −bu, cc, bg, cf, −br}    -   {av, −bb, −ap, −bc, au, aw, −ba, −aq, −bd, at, ax, −az, −ar,        −be, as, ay, −ay, −as, be, ar, az, −ax, −at, bd, aq, ba, −aw,        −au, bc, ap, bb, −av, −av, bb, ap, bc, −au, −aw, ba, aq, bd,        −at, −ax, az, ar, be, −as, −ay, ay, as, −be, −ar, −az, ax, at,        −bd, −aq, −ba, aw, au, −bc, −ap, −bb, av}    -   {bs, −cc, −bi, −cj, bl, bz, −by, −bp, cf, bf, cg, −bo, −bw, by,        bm, −ci, −bh, −cd, br, bt, −cb, −bj, −ck, bk, ca, −bu, −bq, ce,        bg, ch, −bn, −bx, bx, bn, −ch, −bg, −ce, bq, bu, −ca, −bk, ck,        bj, cb, −bt, −br, cd, bh, ci, −bm, −by, bw, bo, −cg, −bf, −cf,        bp, by, −bz, −bl, cj, bi, cc, −bs}    -   {ak, −am, −ai, ao, ah, an, −aj, −al, al, aj, −an, −ah, −ao, ai,        am, −ak, −ak, am, ai, −ao, −ah, −an, aj, al, −al, −aj, an, ah,        ao, −ai, −am, ak, ak, −am, −ai, ao, ah, an, −aj, −al, al, aj,        −an, −ah, −ao, ai, am, −ak, −ak, am, ai, −ao, −ah, −an, aj, al,        −al, −aj, an, ah, ao, −ai, −am, ak}    -   {bt, −bz, −bn, cf, bh, ck, −bi, −ce, bo, by, −bu, −bs, ca, bm,        −cg, −bg, −cj, bj, cd, −bp, −bx, by, br, −cb, −bl, ch, bf, ci,        −bk, −cc, bq, bw, −bw, −bq, cc, bk, −ci, −bf, −ch, bl, cb, −br,        −by, bx, bp, −cd, −bj, cj, bg, cg, −bm, −ca, bs, bu, −by, −bo,        ce, bi, −ck, −bh, −cf, bn, bz, −bt}    -   {aw, −ay, −au, ba, as, −bc, −aq, be, ap, bd, −ar, −bb, at, az,        −av, −ax, ax, av, −az, −at, bb, ar, −bd, −ap, −be, aq, bc, −as,        −ba, au, ay, −aw, −aw, ay, au, −ba, −as, bc, aq, −be, −ap, −bd,        ar, bb, −at, −az, av, ax, −ax, −av, az, at, −bb, −ar, bd, ap,        be, −aq, −bc, as, ba, −au, −ay, aw}    -   {bu, −bw, −bs, by, bq, −ca, −bo, cc, bm, −ce, −bk, cg, bi, −ci,        −bg, ck, bf, cj, −bh, −ch, bj, cf, −bl, −cd, bn, cb, −bp, −bz,        br, bx, −bt, −by, by, bt, −bx, −br, bz, bp, −cb, −bn, cd, bl,        −cf, −bj, ch, bh, −cj, −bf, −ck, bg, ci, −bi, −cg, bk, ce, −bm,        −cc, bo, ca, −bq, −by, bs, bw, −bu}    -   {aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa,        −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa,        aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa,        −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa,        aa, −aa, −aa, aa, aa, −aa, −aa, aa}    -   {by, −bt, −bx, br, bz, −bp, −cb, bn, cd, −bl, −cf, bj, ch, −bh,        −cj, bf, −ck, −bg, ci, bi, −cg, −bk, ce, bm, −cc, −bo, ca, bq,        −by, −bs, bw, bu, −bu, −bw, bs, by, −bq, −ca, bo, cc, −bm, −ce,        bk, cg, −bi, −ci, bg, ck, −bf, cj, bh, −ch, −bj, cf, bl, −cd,        −bn, cb, bp, −bz, −br, bx, bt, −bv}    -   {ax, −av, −az, at, bb, −ar, −bd, ap, −be, −aq, bc, as, −ba, −au,        ay, aw, −aw, −ay, au, ba, −as, −bc, aq, be, −ap, bd, ar, −bb,        −at, az, av, −ax, −ax, av, az, −at, −bb, ar, bd, −ap, be, aq,        −bc, −as, ba, au, −ay, −aw, aw, ay, −au, −ba, as, bc, −aq, −be,        ap, −bd, −ar, bb, at, −az, −av, ax}    -   {bw, −bq, −cc, bk, ci, −bf, ch, bl, −cb, −br, by, bx, −bp, −cd,        bj, cj, −bg, cg, bm, −ca, −bs, bu, by, −bo, −ce, bi, ck, −bh,        cf, bn, −bz, −bt, bt, bz, −bn, −cf, bh, −ck, −bi, ce, bo, −by,        −bu, bs, ca, −bm, −cg, bg, −cj, −bj, cd, bp, −bx, −by, br, cb,        −bl, −ch, bf, −ci, −bk, cc, bq, −bw}    -   {al, −aj, −an, ah, −ao, −ai, am, ak, −ak, −am, ai, ao, −ah, an,        aj, −al, −al, aj, an, −ah, ao, ai, −am, −ak, ak, am, −ai, −ao,        ah, −an, −aj, al, al, −aj, −an, ah, −ao, −ai, am, ak, −ak, −am,        ai, ao, −ah, an, aj, −al, −al, aj, an, −ah, ao, ai, −am, −ak,        ak, am, −ai, −ao, ah, −an, −aj, al}    -   {bx, −bn, −ch, bg, −ce, −bq, bu, ca, −bk, −ck, bj, −cb, −bt, br,        cd, −bh, ci, bm, −by, −bw, bo, cg, −bf, cf, bp, −by, −bz, bl,        cj, −bi, cc, bs, −bs, −cc, bi, −cj, −bl, bz, by, −bp, −cf, bf,        −cg, −bo, bw, by, −bm, −ci, bh, −cd, −br, bt, cb, −bj, ck, bk,        −ca, −bu, bq, ce, −bg, ch, bn, −bx}    -   {ay, −as, −be, ar, −az, −ax, at, bd, −aq, ba, aw, −au, −bc, ap,        −bb, −av, av, bb, −ap, bc, au, −aw, −ba, aq, −bd, −at, ax, az,        −ar, be, as, −ay, −ay, as, be, −ar, az, ax, −at, −bd, aq, −ba,        −aw, au, bc, −ap, bb, av, −av, −bb, ap, −bc, −au, aw, ba, −aq,        bd, at, −ax, −az, ar, −be, −as, ay}    -   {by, −bk, cj, bn, −by, −cb, bh, −cg, −bq, bs, ce, −bf, cd, bt,        −bp, −ch, bi, −ca, −bw, bm, ck, −bl, bx, bz, −bj, ci, bo, −bu,        −cc, bg, −cf, −br, br, cf, −bg, cc, bu, −bo, −ci, bj, −bz, −bx,        bl, −ck, −bm, bw, ca, −bi, ch, bp, −bt, −cd, bf, −ce, −bs, bq,        cg, −bh, cb, by, −bn, −cj, bk, −by}    -   {af, −ad, ag, ae, −ae, −ag, ad, −af, −af, ad, −ag, −ae, ae, ag,        −ad, af, af, −ad, ag, ae, −ae, −ag, ad, −af, −af, ad, −ag, −ae,        ae, ag, −ad, af, af, −ad, ag, ae, −ae, −ag, ad, −af, −af, ad,        −ag, −ae, ae, ag, −ad, af, af, −ad, ag, ae, −ae, −ag, ad, −af,        −af, ad, −ag, −ae, ae, ag, −ad, af}    -   {bz, −bh, ce, bu, −bm, cj, bp, −br, −ch, bk, −bw, −cc, bf, −cb,        −bx, bj, −cg, −bs, bo, ck, −bn, bt, cf, −bi, by, ca, −bg, cd,        by, −bl, ci, bq, −bq, −ci, bl, −by, −cd, bg, −ca, −by, bi, −cf,        −bt, bn, −ck, −bo, bs, cg, −bj, bx, cb, −bf, cc, bw, −bk, ch,        br, −bp, −cj, bm, −bu, −ce, bh, −bz}    -   {az, −ap, ba, ay, −aq, bb, ax, −ar, bc, aw, −as, bd, av, −at,        be, au, −au, −be, at, −av, −bd, as, −aw, −bc, ar, −ax, −bb, aq,        −ay, −ba, ap, −az, −az, ap, −ba, −ay, aq, −bb, −ax, ar, −bc,        −aw, as, −bd, −av, at, −be, −au, au, be, −at, av, bd, −as, aw,        bc, −ar, ax, bb, −aq, ay, ba, −ap, az}    -   {ca, −bf, bz, cb, −bg, by, cc, −bh, bx, cd, −bi, bw, ce, −bj,        by, cf, −bk, bu, cg, −bl, bt, ch, −bm, bs, ci, −bn, br, cj, −bo,        bq, ck, −bp, bp, −ck, −bq, bo, −cj, −br, bn, −ci, −bs, bm, −ch,        −bt, bl, −cg, −bu, bk, −cf, −by, bj, −ce, −bw, bi, −cd, −bx, bh,        −cc, −by, bg, −cb, −bz, bf, −ca}    -   {am, −ah, al, an, −ai, ak, ao, −aj, aj, −ao, −ak, ai, −an, −al,        ah, −am, −am, ah, −al, −an, ai, −ak, −ao, aj, −aj, ao, ak, −ai,        an, al, −ah, am, am, −ah, al, an, −ai, ak, ao, −aj, aj, −ao,        −ak, ai, −an, −al, ah, −am, −am, ah, −al, −an, ai, −ak, −ao, aj,        −aj, ao, ak, −ai, an, al, −ah, am}    -   {cb, −bi, bu, ci, −bp, bn, −cg, −bw, bg, −bz, −cd, bk, −bs, −ck,        br, −bl, ce, by, −bf, bx, cf, −bm, bq, −cj, −bt, bj, −cc, −ca,        bh, −by, −ch, bo, −bo, ch, by, −bh, ca, cc, −bj, bt, cj, −bq,        bm, −cf, −bx, bf, −by, −ce, bl, −br, ck, bs, −bk, cd, bz, −bg,        bw, cg, −bn, bp, −ci, −bu, bi, −cb}    -   {ba, −ar, av, −be, −aw, aq, −az, −bb, as, −au, bd, ax, −ap, ay,        bc, −at, at, −bc, −ay, ap, −ax, −bd, au, −as, bb, az, −aq, aw,        be, −av, ar, −ba, −ba, ar, −av, be, aw, −aq, az, bb, −as, au,        −bd, −ax, ap, −ay, −bc, at, −at, bc, ay, −ap, ax, bd, −au, as,        −bb, −az, aq, −aw, −be, av, −ar, ba}    -   {cc, −bl, bp, −cg, −by, bh, −bt, ck, bu, −bg, bx, ch, −bq, bk,        −cb, −cd, bm, −bo, cf, bz, −bi, bs, −cj, −by, bf, −bw, −ci, br,        −bj, ca, ce, −bn, bn, −ce, −ca, bj, −br, ci, bw, −bf, by, cj,        −bs, bi, −bz, −cf, bo, −bm, cd, cb, −bk, bq, −ch, −bx, bg, −bu,        −ck, bt, −bh, by, cg, −bp, bl, −cc}    -   {ac, −ab, ab, −ac, −ac, ab, −ab, ac, ac, −ab, ab, −ac, −ac, ab,        −ab, ac, ac, −ab, ab, −ac, −ac, ab, −ab, ac, ac, −ab, ab, −ac,        −ac, ab, −ab, ac, ac, −ab, ab, −ac, −ac, ab, −ab, ac, ac, −ab,        ab, −ac, −ac, ab, −ab, ac, ac, −ab, ab, −ac, −ac, ab, −ab, ac,        ac, −ab, ab, −ac, −ac, ab, −ab, ac}    -   {cd, −bo, bk, −bz, −ch, bs, −bg, by, −ck, −bw, bh, −br, cg, ca,        −bl, bn, −cc, −ce, bp, −bj, by, ci, −bt, bf, −bu, cj, bx, −bi,        bq, −cf, −cb, bm, −bm, cb, cf, −bq, bi, −bx, −cj, bu, −bf, bt,        −ci, −by, bj, −bp, ce, cc, −bn, bl, −ca, −cg, br, −bh, bw, ck,        −by, bg, −bs, ch, bz, −bk, bo, −cd}    -   {bb, −au, aq, −ax, be, ay, −ar, at, −ba, −bc, av, −ap, aw, −bd,        −az, as, −as, az, bd, −aw, ap, −av, bc, ba, −at, ar, −ay, −be,        ax, −aq, au, −bb, −bb, au, −aq, ax, −be, −ay, ar, −at, ba, bc,        −av, ap, −aw, bd, az, −as, as, −az, −bd, aw, −ap, av, −bc, −ba,        at, −ar, ay, be, −ax, aq, −au, bb}    -   {ce, −br, bf, −bs, cf, cd, −bq, bg, −bt, cg, cc, −bp, bh, −bu,        ch, cb, −bo, bi, −by, ci, ca, −bn, bj, −bw, cj, bz, −bm, bk,        −bx, ck, by, −bl, bl, −by, −ck, bx, −bk, bm, −bz, −cj, bw, −bj,        bn, −ca, −ci, by, −bi, bo, −cb, −ch, bu, −bh, bp, −cc, −cg, bt,        −bg, bq, −cd, −cf, bs, −bf, br, −ce}    -   {an, −ak, ah, −aj, am, ao, −al, ai, −ai, al, −ao, −am, aj, −ah,        ak, −an, −an, ak, −ah, aj, −am, −ao, al, −ai, ai, −al, ao, am,        −aj, ah, −ak, an, an, −ak, ah, −aj, am, ao, −al, ai, −ai, al,        −ao, −am, aj, −ah, ak, −an, −an, ak, −ah, aj, −am, −ao, al, −ai,        ai, −al, ao, am, −aj, ah, −ak, an}    -   {cf, −bu, bj, −bl, bw, −ch, −cd, bs, −bh, bn, −by, cj, cb, −bq,        bf, −bp, ca, ck, −bz, bo, −bg, br, −cc, −ci, bx, −bm, bi, −bt,        ce, cg, −by, bk, −bk, by, −cg, −ce, bt, −bi, bm, −bx, ci, cc,        −br, bg, −bo, bz, −ck, −ca, bp, −bf, bq, −cb, −cj, by, −bn, bh,        −bs, cd, ch, −bw, bl, −bj, bu, −cf}    -   {bc, −ax, as, −aq, av, −ba, −be, az, −au, ap, −at, ay, −bd, −bb,        aw, −ar, ar, −aw, bb, bd, −ay, at, −ap, au, −az, be, ba, −av,        aq, −as, ax, −bc, −bc, ax, −as, aq, −av, ba, be, −az, au, −ap,        at, −ay, bd, bb, −aw, ar, −ar, aw, −bb, −bd, ay, −at, ap, −au,        az, −be, −ba, av, −aq, as, −ax, bc}    -   {cg, −bx, bo, −bf, bn, −bw, cf, ch, −by, bp, −bg, bm, −by, ce,        ci, −bz, bq, −bh, bl, −bu, cd, cj, −ca, br, −bi, bk, −bt, cc,        ck, −cb, bs, −bj, bj, −bs, cb, −ck, −cc, bt, −bk, bi, −br, ca,        −cj, −cd, bu, −bl, bh, −bq, bz, −ci, −ce, by, −bm, bg, −bp, by,        −ch, −cf, bw, −bn, bf, −bo, bx, −cg}    -   {ag, −af, ae, −ad, ad, −ae, af, −ag, −ag, af, −ae, ad, −ad, ae,        −af, ag, ag, −af, ae, −ad, ad, −ae, af, −ag, −ag, af, −ae, ad,        −ad, ae, −af, ag, ag, −af, ae, −ad, ad, −ae, af, −ag, −ag, af,        −ae, ad, −ad, ae, −af, ag, ag, −af, ae, −ad, ad, −ae, af, −ag,        −ag, af, −ae, ad, −ad, ae, −af, ag}    -   {ch, −ca, bt, −bm, bf, −bl, bs, −bz, cg, ci, −cb, bu, −bn, bg,        −bk, br, −by, cf, cj, −cc, by, −bo, bh, −bj, bq, −bx, ce, ck,        −cd, bw, −bp, bi, −bi, bp, −bw, cd, −ck, −ce, bx, −bq, bj, −bh,        bo, −by, cc, −cj, −cf, by, −br, bk, −bg, bn, −bu, cb, −ci, −cg,        bz, −bs, bl, −bf, bm, −bt, ca, −ch}    -   {bd, −ba, ax, −au, ar, −ap, as, −av, ay, −bb, be, bc, −az, aw,        −at, aq, −aq, at, −aw, az, −bc, −be, bb, −ay, av, −as, ap, −ar,        au, −ax, ba, −bd, −bd, ba, −ax, au, −ar, ap, −as, av, −ay, bb,        −be, −bc, az, −aw, at, −aq, aq, −at, aw, −az, bc, be, −bb, ay,        −av, as, −ap, ar, −au, ax, −ba, bd}    -   {ci, −cd, by, −bt, bo, −bj, bf, −bk, bp, −bu, bz, −ce, cj, ch,        −cc, bx, −bs, bn, −bi, bg, −bl, bq, −by, ca, −cf, ck, cg, −cb,        bw, −br, bm, −bh, bh, −bm, br, −bw, cb, −cg, −ck, cf, −ca, by,        −bq, bl, −bg, bi, −bn, bs, −bx, cc, −ch, −cj, ce, −bz, bu, −bp,        bk, −bf, bj, −bo, bt, −by, cd, −ci}    -   {ao, −an, am, −al, ak, −aj, ai, −ah, ah, −ai, aj, −ak, al, −am,        an, −ao, −ao, an, −am, al, −ak, aj, −ai, ah, −ah, ai, −aj, ak,        −al, am, −an, ao, ao, −an, am, −al, ak, −aj, ai, −ah, ah, −ai,        aj, −ak, al, −am, an, −ao, −ao, an, −am, al, −ak, aj, −ai, ah,        −ah, ai, −aj, ak, −al, am, −an, ao}    -   {cj, −cg, cd, −ca, bx, −bu, br, −bo, bl, −bi, bf, −bh, bk, −bn,        bq, −bt, bw, −bz, cc, −cf, ci, ck, −ch, ce, −cb, by, −by, bs,        −bp, bm, −bj, bg, −bg, bj, −bm, bp, −bs, by, −by, cb, −ce, ch,        −ck, −ci, cf, −cc, bz, −bw, bt, −bq, bn, −bk, bh, −bf, bi, −bl,        bo, −br, bu, −bx, ca, −cd, cg, −el}    -   {be, −bd, bc, −bb, ba, −az, ay, −ax, aw, −av, au, −at, as, −ar,        aq, −ap, ap, −aq, ar, −as, at, −au, av, −aw, ax, −ay, az, −ba,        bb, −bc, bd, −be, −be, bd, −bc, bb, −ba, az, −ay, ax, −aw, av,        −au, at, −as, ar, −aq, ap, −ap, aq, −ar, as, −at, au, −av, aw,        −ax, ay, −az, ba, −bb, bc, −bd, be}    -   {ck, −cj, ci, −ch, cg, −cf, ce, −cd, cc, −cb, ca, −bz, by, −bx,        bw, −by, bu, −bt, bs, −br, bq, −bp, bo, −bn, bm, −bl, bk, −bj,        bi, −bh, bg, −bf, bf, −bg, bh, −bi, bj, −bk, bl, −bm, bn, −bo,        bp, −bq, br, −bs, bt, −bu, by, −bw, bx, −by, bz, −ca, cb, −cc,        cd, −ce, cf, −cg, ch, −ci, cj, −ck}

An exemplary 128-point DCT-2 core defined in VVC is shown below, where{aa, ab, ac, ad, ae, af, ag, ah, ai, aj, ak, al, am, an, ao, ap, aq, ar,as, at, au, av, aw, ax, ay, az, ba, bb, bc, bd, be, bf, bg, bh, bi, bj,bk, bl, bm, bn, bo, bp, bq, br, bs, bt, bu, by, bw, bx, by, bz, ca, cb,cc, cd, ce, cf, cg, ch, ci, cj, ck, cl, cm, cn, co, cp, cq, cr, cs, ct,cu, cv, cw, cx, cy, cz, da, db, dc, dd, de, df, dg, dh, di, dj, dk, dl,dm, dn, do, dp, dq, dr, ds, dt, du, dv, dw, dx, dy, dz, ea, eb, ec, ed,ee, ef, eg, eh, ei, ej, ek, el, em, en, eo, ep, eq, er, es, et, eu, ev,ew}={256, 334, 139, 355, 301, 201, 71, 360, 346, 319, 280, 230, 171,105, 35, 362, 358, 351, 341, 327, 311, 291, 268, 243, 216, 186, 155,122, 88, 53, 18, 362, 361, 359, 357, 353, 349, 344, 338, 331, 323, 315,306, 296, 285, 274, 262, 250, 236, 223, 208, 194, 178, 163, 147, 130,114, 97, 79, 62, 44, 27, 9, 362, 362, 361, 361, 360, 359, 357, 356, 354,352, 350, 348, 345, 342, 339, 336, 333, 329, 325, 321, 317, 313, 308,303, 299, 293, 288, 283, 277, 271, 265, 259, 253, 246, 240, 233, 226,219, 212, 205, 197, 190, 182, 175, 167, 159, 151, 143, 134, 126, 118,109, 101, 92, 84, 75, 66, 58, 49, 40, 31, 22, 13, 4}.

The 128-point DCT-2 core is shown below, only the left half of the128×128 transform core is described, the right half can be derivedaccording to the symmetry/anti-symmetry characteristics of DCT-2:

-   -   {aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa,        aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa,        aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa,        aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa}    -   {cl, cm, cn, co, cp, cq, cr, cs, ct, cu, cv, cw, cx, cy, cz, da,        db, dc, dd, de, df, dg, dh, di, dj, dk, dl, dm, dn, d_, dp, dq,        dr, ds, dt, du, dv, dw, dx, dy, dz, ea, eb, ec, ed, ee, ef, eg,        eh, ei, ej, ek, el, em, en, eo, ep, eq, er, es, et, eu, ev, ew}    -   {bf, bg, bh, bi, bj, bk, bl, bm, bn, bo, bp, bq, br, bs, bt, bu,        by, bw, bx, by, bz, ca, cb, cc, cd, ce, cf, cg, ch, ci, cj, ck,        −ck, −cj, −ci, −ch, −cg, −cf, −ce, −cd, −cc, −cb, −ca, −bz, −by,        −bx, −bw, −by, −bu, −bt, −bs, −br, −bq, −bp, −bo, −bn, −bm, −bl,        −bk, −bj, −bi, −bh, −bg, −bf}    -   {cm, cp, cs, cv, cy, db, de, dh, dk, dn, dq, dt, dw, dz, ec, ef,        ei, el, eo, er, eu, −ew, −et, −eq, −en, −ek, −eh, −ee, −eb, −dy,        −dv, −ds, −dp, −dm, −dj, −dg, −dd, −da, −cx, −cu, −cr, −co, −cl,        −cn, −cq, −ct, −cw, −cz, −dc, −df, −di, −dl, −d_, −dr, −du, −dx,        −ea, −ed, −eg, −ej, −em, −ep, −es, −ev}    -   {ap, aq, ar, as, at, au, av, aw, ax, ay, az, ba, bb, bc, bd, be,        −be, −bd, −bc, −bb, −ba, −az, −ay, −ax, −aw, −av, −au, −at, −as,        −ar, −aq, −ap, −ap, −aq, −ar, −as, −at, −au, −av, −aw, −ax, −ay,        −az, −ba, −bb, −bc, −bd, −be, be, bd, bc, bb, ba, az, ay, ax,        aw, av, au, at, as, ar, aq, ap}    -   {cn, cs, cx, dc, dh, dm, dr, dw, eb, eg, el, eq, ev, −et, −eo,        −ej, −ee, −dz, −du, −dp, −dk, −df, −da, −cv, −cq, −cl, −cp, −cu,        −cz, −de, −dj, −d_, −dt, −dy, −ed, −ei, −en, −es, ew, er, em,        eh, ec, dx, ds, dn, di, dd, cy, ct, co, cm, cr, cw, db, dg, dl,        dq, dv, ea, ef, ek, ep, eu}    -   {bg, bj, bm, bp, bs, by, by, cb, ce, ch, ck, −ci, −cf, −cc, −bz,        −bw, −bt, −bq, −bn, −bk, −bh, −bf, −bi, −bl, −bo, −br, −bu, −bx,        −ca, −cd, −cg, −cj, cj, cg, cd, ca, bx, bu, br, bo, bl, bi, bf,        bh, bk, bn, bq, bt, bw, bz, cc, cf, ci, −ck, −ch, −ce, −cb, −by,        −by, −bs, −bp, −bm, −bj, −bg}    -   {co, cv, dc, dj, dq, dx, ee, el, es, −eu, −en, −eg, −dz, −ds,        −dl, −de, −cx, −cq, −cm, −ct, −da, −dh, −d_, −dv, −ec, −ej, −eq,        ew, ep, ei, eb, du, dn, dg, cz, cs, cl, cr, cy, df, dm, dt, ea,        eh, eo, ev, −er, −ek, −ed, −dw, −dp, −di, −db, −cu, −cn, −cp,        −cw, −dd, −dk, −dr, −dy, −ef, −em, −et}    -   {ah, ai, aj, ak, al, am, an, ao, −ao, −an, −am, −al, −ak, −aj,        −ai, −ah, −ah, −ai, −aj, −ak, −al, −am, −an, −ao, ao, an, am,        al, ak, aj, ai, ah, ah, ai, aj, ak, al, am, an, ao, −ao, −an,        −am, −al, −ak, −aj, −ai, −ah, −ah, −ai, −aj, −ak, −al, −am, −an,        −ao, ao, an, am, al, ak, aj, ai, ah}    -   {cp, cy, dh, dq, dz, ei, er, −et, −ek, −eb, −ds, −dj, −da, −cr,        −cn, −cw, −df, −d_, −dx, −eg, −ep, ev, em, ed, du, dl, dc, ct,        cl, cu, dd, dm, dv, ee, en, ew, −eo, −ef, −dw, −dn, −de, −cv,        −cm, −cs, −db, −dk, −dt, −ec, −el, −eu, eq, eh, dy, dp, dg, cx,        co, cq, cz, di, dr, ea, ej, es}    -   {bh, bm, br, bw, cb, cg, −ck, −cf, −ca, −by, −bq, −bl, −bg, −bi,        −bn, −bs, −bx, −cc, −ch, cj, ce, bz, bu, bp, bk, bf, bj, bo, bt,        by, cd, ci, −ci, −cd, −by, −bt, −bo, −bj, −bf, −bk, −bp, −bu,        −bz, −ce, −cj, ch, cc, bx, bs, bn, bi, bg, bl, bq, by, ca, cf,        ck, −cg, −cb, −bw, −br, −bm, −bh}    -   {cq, db, dm, dx, ei, et, −ep, −ee, −dt, −di, −cx, −cm, −cu, −df,        −dq, −eb, −em, ew, el, ea, dp, de, ct, cn, cy, dj, du, ef, eq,        −es, −eh, −dw, −dl, −da, −cp, −cr, −dc, −dn, −dy, −ej, −eu, eo,        ed, ds, dh, cw, cl, cv, dg, dr, ec, en, −ev, −ek, −dz, −d_, −dd,        −cs, −co, −cz, −dk, −dv, −eg, −er}    -   {aq, at, aw, az, bc, −be, −bb, −ay, −av, −as, −ap, −ar, −au,        −ax, −ba, −bd, bd, ba, ax, au, ar, ap, as, av, ay, bb, be, −bc,        −az, −aw, −at, −aq, −aq, −at, −aw, −az, −bc, be, bb, ay, av, as,        ap, ar, au, ax, ba, bd, −bd, −ba, −ax, −au, −ar, −ap, −as, −av,        −ay, −bb, −be, bc, az, aw, at, aq}    -   {cr, de, dr, ee, er, −ep, −ec, −dp, −dc, −cp, −ct, −dg, −dt,        −eg, −et, en, ea, dn, da, cn, cv, di, dv, ei, ev, −el, −dy, −dl,        −cy, −cl, −cx, −dk, −dx, −ek, ew, ej, dw, dj, cw, cm, cz, dm,        dz, em, −eu, −eh, −du, −dh, −cu, −co, −db, −d_, −eb, −eo, es,        ef, ds, df, cs, cq, dd, dq, ed, eq}    -   {bi, bp, bw, cd, ck, −ce, −bx, −bq, −bj, −bh, −bo, −by, −cc,        −cj, cf, by, br, bk, bg, bn, bu, cb, ci, −cg, −bz, −bs, −bl,        −bf, −bm, −bt, −ca, −ch, ch, ca, bt, bm, bf, bl, bs, bz, cg,        −ci, −cb, −bu, −bn, −bg, −bk, −br, −by, −cf, cj, cc, by, bo, bh,        bj, bq, bx, ce, −ck, −cd, −bw, −bp, −bi}    -   {cs, dh, dw, el, −et, −ee, −dp, −da, −cl, −cz, −d_, −ed, −es,        em, dx, di, ct, cr, dg, dv, ek, −eu, −ef, −dq, −db, −cm, −cy,        −dn, −ec, −er, en, dy, dj, cu, cq, df, du, ej, −ev, −eg, −dr,        −dc, −cn, −cx, −dm, −eb, −eq, eo, dz, dk, cv, cp, de, dt, ei,        −ew, −eh, −ds, −dd, −co, −cw, −dl, −ea, −ep}    -   {ad, ae, af, ag, −ag, −af, −ae, −ad, −ad, −ae, −af, −ag, ag, af,        ae, ad, ad, ae, af, ag, −ag, −af, −ae, −ad, −ad, −ae, −af, −ag,        ag, af, ae, ad, ad, ae, af, ag, −ag, −af, −ae, −ad, −ad, −ae,        −af, −ag, ag, af, ae, ad, ad, ae, af, ag, −ag, −af, −ae, −ad,        −ad, −ae, −af, −ag, ag, af, ae, ad}    -   {ct, dk, eb, es, −ek, −dt, −dc, −cl, −db, −ds, −ej, et, ec, dl,        cu, cs, dj, ea, er, −el, −du, −dd, −cm, −da, −dr, −ei, eu, ed,        dm, cv, cr, di, dz, eq, −em, −dv, −de, −cn, −cz, −dq, −eh, ev,        ee, dn, cw, cq, dh, dy, ep, −en, −dw, −df, −co, −dp, −eg, ew,        ef, d_, cx, cp, dg, dx, eo}{bj, bs, cb, ck, −cc, −bt, −bk, −bi,        −br, −ca, −cj, cd, bu, bl, bh, bq, bz, ci, −ce, −by, −bm, −bg,        −bp, −by, −ch, cf, bw, bn, bf, bo, bx, cg, −cg, −bx, −bo, −bf,        −bn, −bw, −cf, ch, by, bp, bg, bm, by, ce, −ci, −bz, −bq, −bh,        −bl, −bu, −cd, cj, ca, br, bi, bk, bt, cc, −ck, −cb, −bs, −bj}    -   {cu, dn, eg, −eu, −eb, −di, −cp, −cz, −ds, −el, ep, dw, dd, cl,        de, dx, eq, −ek, −dr, −cy, −cq, −dj, −ec, −ev, ef, dm, ct, cv,        d_, eh, −et, −ea, −dh, −co, −da, −dt, −em, eo, dv, dc, cm, df,        dy, er, −ej, −dq, −cx, −cr, −dk, −ed, −ew, ee, dl, cs, cw, dp,        ei, −es, −dz, −dg, −cn, −db, −du, −en}    -   {ar, aw, bb, −bd, −ay, −at, −ap, −au, −az, −be, ba, av, aq, as,        ax, bc, −bc, −ax, −as, −aq, −av, −ba, be, az, au, ap, at, ay,        bd, −bb, −aw, −ar, −ar, −aw, −bb, bd, ay, at, ap, au, az, be,        −ba, −av, −aq, −as, −ax, −bc, bc, ax, as, aq, av, ba, −be, −az,        −au, −ap, −at, −ay, −bd, bb, aw, ar}    -   {cv, dq, el, −en, −ds, −cx, −ct, −d_, −ej, ep, du, cz, cr, dm,        eh, −er, −dw, −db, −cp, −dk, −ef, et, dy, dd, cn, di, ed, −ev,        −ea, −df, −cl, −dg, −eb, −ew, ec, dh, cm, de, dz, eu, −ee, −dj,        −co, −dc, −dx, −es, eg, dl, cq, da, dv, eq, −ei, −dn, −cs, −cy,        −dt, −eo, ek, dp, cu, cw, dr, em}    -   {bk, by, cg, −ce, −bt, −bi, −bm, −bx, −ci, cc, br, bg, bo, bz,        ck, −ca, −bp, −bf, −bq, −cb, cj, by, bn, bh, bs, cd, −ch, −bw,        −bl, −bj, −bu, −cf, cf, bu, bj, bl, bw, ch, −cd, −bs, −bh, −bn,        −by, −cj, cb, bq, bf, bp, ca, −ck, −bz, −bo, −bg, −br, −cc, ci,        bx, bm, bi, bt, ce, −cg, −by, −bk}    -   {cw, dt, eq, −eg, −dj, −cm, −dg, −ed, et, dw, cz, ct, dq, en,        −ej, −dm, −cp, −dd, −ea, ew, dz, dc, cq, dn, ek, −em, −dp, −cs,        −da, −dx, −eu, ec, df, cn, dk, eh, −ep, −ds, −cv, −cx, −du, −er,        ef, di, cl, dh, ee, −es, −dv, −cy, −cu, −dr, −eo, ei, dl, co,        de, eb, −ev, −dy, −db, −cr, −d_, −el}    -   {ai, al, ao, −am, −aj, −ah, −ak, −an, an, ak, ah, aj, am, −ao,        −al, −ai, −ai, −al, −ao, am, aj, ah, ak, an, −an, −ak, −ah, −aj,        −am, ao, al, ai, ai, al, ao, −am, −aj, −ah, −ak, −an, an, ak,        ah, aj, am, −ao, −al, −ai, −ai, −al, −ao, am, aj, ah, ak, an,        −an, −ak, −ah, −aj, −am, ao, al, ai}    -   {cx, dw, ev, −dz, −da, −cu, −dt, −es, ec, dd, cr, dq, ep, −ef,        −dg, −co, −dn, −em, ei, dj, cl, dk, ej, −el, −dm, −cn, −dh, −eg,        eo, dp, cq, de, ed, −er, −ds, −ct, −db, −ea, eu, dv, cw, cy, dx,        ew, −dy, −cz, −cv, −du, −et, eb, dc, cs, dr, eq, −ee, −df, −cp,        −d_, −en, eh, di, cm, dl, ek}    -   {bl, by, −ck, −bx, −bk, −bm, −bz, cj, bw, bj, bn, ca, −ci, −by,        −bi, −bo, −cb, ch, bu, bh, bp, cc, −cg, −bt, −bg, −bq, −cd, cf,        bs, bf, br, ce, −ce, −br, −bf, −bs, −cf, cd, bq, bg, bt, cg,        −cc, −bp, −bh, −bu, −ch, cb, bo, bi, by, ci, −ca, −bn, −bj, −bw,        −cj, bz, bm, bk, bx, ck, −by, −bl}    -   {cy, dz, −et, −ds, −cr, −df, −eg, em, dl, cl, dm, en, −ef, −de,        −cs, −dt, −eu, dy, cx, cz, ea, −es, −dr, −cq, −dg, −eh, el, dk,        cm, dn, eo, −ee, −dd, −ct, −du, −ev, dx, cw, da, eb, −er, −dq,        −cp, −dh, −ei, ek, dj, cn, d_, ep, −ed, −dc, −cu, −dv, −ew, dw,        cv, db, ec, −eq, −dp, −co, −di, −ej}    -   {as, az, −bd, −aw, −ap, −av, −bc, ba, at, ar, ay, −be, −ax, −aq,        −au, −bb, bb, au, aq, ax, be, −ay, −ar, −at, −ba, bc, av, ap,        aw, bd, −az, −as, −as, −az, bd, aw, ap, av, bc, −ba, −at, −ar,        −ay, be, ax, aq, au, bb, −bb, −au, −aq, −ax, −be, ay, ar, at,        ba, −bc, −av, −ap, −aw, −bd, az, as}    -   {cz, ec, −eo, −dl, −cn, −dq, −et, dx, cu, de, eh, −ej, −dg, −cs,        −dv, ev, ds, cp, dj, em, −ee, −db, −cx, −ea, eq, dn, cl, d_, er,        −dz, −cw, −dc, −ef, el, di, cq, dt, ew, −du, −cr, −dh, −ek, eg,        dd, cv, dy, −es, −dp, −cm, −dm, −ep, eb, cy, da, ed, −en, −dk,        −co, −dr, −eu, dw, ct, df, ei}    -   {bm, cb, −cf, −bq, −bi, −bx, cj, bu, bf, bt, ci, −by, −bj, −bp,        −ce, cc, bn, bl, ca, −cg, −br, −bh, −bw, ck, by, bg, bs, ch,        −bz, −bk, −bo, −cd, cd, bo, bk, bz, −ch, −bs, −bg, −by, −ck, bw,        bh, br, cg, −ca, −bl, −bn, −cc, ce, bp, bj, by, −ci, −bt, −bf,        −bu, −cj, bx, bi, bq, cf, −cb, −bm}    -   {da, ef, −ej, −de, −cw, −eb, en, di, cs, dx, −er, −dm, −co, −dt,        ev, dq, cl, dp, eu, −du, −cp, −dl, −eq, dy, ct, dh, em, −ec,        −cx, −dd, −ei, eg, db, cz, ee, −ek, −df, −cv, −ea, eo, dj, cr,        dw, −es, −dn, −cn, −ds, ew, dr, cm, d_, et, −dv, −cq, −dk, −ep,        dz, cu, dg, el, −ed, −dc, −eh}    -   {ab, ac, −ac, −ab, −ab, −ac, ac, ab, ab, ac, −ac, −ab, −ab, −ac,        ac, ab, ab, ac, −ac, −ab, −ab, −ac, ac, ab, ab, ac, −ac, −ab,        −ab, −ac, ac, ab, ab, ac, −ac, −ab, −ab, −ac, ac, ab, ab, ac,        −ac, −ab, −ab, −ac, ac, ab, ab, ac, −ac, −ab, −ab, −ac, ac, ab,        ab, ac, −ac, −ab, −ab, −ac, ac, ab}    -   {db, ei, −ee, −cx, −df, −em, ea, ct, dj, eq, −dw, −cp, −dn, −eu,        ds, cl, dr, −ev, −d_, −co, −dv, er, dk, cs, dz, −en, −dg, −cw,        −ed, ej, dc, da, eh, −ef, −cy, −de, −el, eb, cu, di, ep, −dx,        −cq, −dm, −et, dt, cm, dq, −ew, −dp, −cn, −du, es, dl, cr, dy,        −eo, −dh, −cv, −ec, ek, dd, cz, eg}    -   {bn, ce, −ca, −bj, −br, −ci, bw, bf, by, −cj, −bs, −bi, −bz, cf,        bo, bm, cd, −cb, −bk, −bq, −ch, bx, bg, bu, −ck, −bt, −bh, −by,        cg, bp, bl, cc, −cc, −bl, −bp, −cg, by, bh, bt, ck, −bu, −bg,        −bx, ch, bq, bk, cb, −cd, −bm, −bo, −cf, bz, bi, bs, cj, −by,        −bf, −bw, ci, br, bj, ca, −ce, −bn}    -   {dc, el, −dz, −cq, −d_, ew, dn, cr, ea, −ek, −db, −dd, −em, dy,        cp, dp, −ev, −dm, −cs, −eb, ej, da, de, en, −dx, −co, −dq, eu,        dl, ct, ec, −ei, −cz, −df, −eo, dw, cn, dr, −et, −dk, −cu, −ed,        eh, cy, dg, ep, −dv, −cm, −ds, es, dj, cv, ee, −eg, −cx, −dh,        −eq, du, cl, dt, −er, −di, −cw, −ef}    -   {at, bc, −ay, −ap, −ax, bd, au, as, bb, −az, −aq, −aw, be, av,        ar, ba, −ba, −ar, −av, −be, aw, aq, az, −bb, −as, −au, −bd, ax,        ap, ay, −bc, −at, −at, −bc, ay, ap, ax, −bd, −au, −as, −bb, az,        aq, aw, −be, −av, −ar, −ba, ba, ar, av, be, −aw, −aq, −az, bb,        as, au, bd, −ax, −ap, −ay, bc, at}    -   {dd, eo, −du, −cm, −dx, el, da, dg, er, −dr, −cp, −ea, ei, cx,        dj, eu, −d_, −cs, −ed, ef, cu, dm, −ew, −dl, −cv, −eg, ec, cr,        dp, −et, −di, −cy, −ej, dz, co, ds, −eq, −df, −db, −em, dw, cl,        dv, −en, −dc, −de, −ep, dt, cn, dy, −ek, −cz, −dh, −es, dq, cq,        eb, −eh, −cw, −dk, −ev, dn, ct, ee}    -   {bo, ch, −by, −bh, −ca, cc, bj, bt, −cj, −bq, −bm, −cf, bx, bf,        by, −ce, −bl, −br, −ck, bs, bk, cd, −bz, −bg, −bw, cg, bn, bp,        ci, −bu, −bi, −cb, cb, bi, bu, −ci, −bp, −bn, −cg, bw, bg, bz,        −cd, −bk, −bs, ck, br, bl, ce, −by, −bf, −bx, cf, bm, bq, cj,        −bt, −bj, −cc, ca, bh, by, −ch, −bo}    -   {de, er, −dp, −ct, −eg, ea, cn, dv, −el, −cy, −dk, ew, dj, cz,        em, −du, −co, −eb, ef, cs, dq, −eq, −dd, −df, −es, d_, cu, eh,        −dz, −cm, −dw, ek, cx, dl, −ev, −di, −da, −en, dt, cp, ec, −ee,        −cr, −dr, ep, dc, dg, et, −dn, −cv, −ei, dy, cl, dx, −ej, −cw,        −dm, eu, dh, db, eo, −ds, −cq, −ed}    -   {aj, ao, −ak, −ai, −an, al, ah, am, −am, −ah, −al, an, ai, ak,        −ao, −aj, −aj, −ao, ak, ai, an, −al, −ah, −am, am, ah, al, −an,        −ai, −ak, ao, aj, aj, ao, −ak, −ai, −an, al, ah, am, −am, −ah,        −al, an, ai, ak, −ao, −aj, −aj, −ao, ak, ai, an, −al, −ah, −am,        am, ah, al, −an, −ai, −ak, ao, aj}    -   {df, eu, −dk, −da, −ep, dp, cv, ek, −du, −cq, −ef, dz, cl, ea,        −ee, −cp, −dv, ej, cu, dq, −eo, −cz, −dl, et, de, dg, ev, −dj,        −db, −eq, d_, cw, el, −dt, −cr, −eg, dy, cm, eb, −ed, −co, −dw,        ei, ct, dr, −en, −cy, −dm, es, dd, dh, ew, −di, −dc, −er, dn,        cx, em, −ds, −cs, −eh, dx, cn, ec}    -   {bp, ck, −bq, −bo, −cj, br, bn, ci, −bs, −bm, −ch, bt, bl, cg,        −bu, −bk, −cf, by, bj, ce, −bw, −bi, −cd, bx, bh, cc, −by, −bg,        −cb, bz, bf, ca, −ca, −bf, −bz, cb, bg, by, −cc, −bh, −bx, cd,        bi, bw, −ce, −bj, −by, cf, bk, bu, −cg, −bl, −bt, ch, bm, bs,        −ci, −bn, −br, cj, bo, bq, −ck, −bp}    -   {dg, −ew, −df, −dh, ev, de, di, −eu, −dd, −dj, et, dc, dk, −es,        −db, −dl, er, da, dm, −eq, −cz, −dn, ep, cy, d_, −eo, −cx, −dp,        en, cw, dq, −em, −cv, −dr, el, cu, ds, −ek, −ct, −dt, ej, cs,        du, −ei, −cr, −dv, eh, cq, dw, −eg, −cp, −dx, ef, co, dy, −ee,        −cn, −dz, ed, cm, ea, −ec, −cl, −eb}    -   {au, −be, −at, −av, bd, as, aw, −bc, −ar, −ax, bb, aq, ay, −ba,        −ap, −az, az, ap, ba, −ay, −aq, −bb, ax, ar, bc, −aw, −as, −bd,        av, at, be, −au, −au, be, at, av, −bd, −as, −aw, bc, ar, ax,        −bb, −aq, −ay, ba, ap, az, −az, −ap, −ba, ay, aq, bb, −ax, −ar,        −bc, aw, as, bd, −av, −at, −be, au}    -   {dh, −et, −da, −d_, em, ct, dv, −ef, −cm, −ec, dy, cq, ej, −dr,        −cx, −eq, dk, de, −ew, −dd, −dl, ep, cw, ds, −ei, −cp, −dz, eb,        cn, eg, −du, −cu, −en, dn, db, eu, −dg, −di, es, cz, dp, −el,        −cs, −dw, ee, cl, ed, −dx, −cr, −ek, dq, cy, er, −dj, −df, ev,        dc, dm, −eo, −cv, −dt, eh, co, ea}    -   {bq, −ci, −bl, −by, cd, bg, ca, −by, −bi, −cf, bt, bn, ck, −bo,        −bs, cg, bj, bx, −cb, −bf, −cc, bw, bk, ch, −br, −bp, cj, bm,        bu, −ce, −bh, −bz, bz, bh, ce, −bu, −bm, −cj, bp, br, −ch, −bk,        −bw, cc, bf, cb, −bx, −bj, −cg, bs, bo, −ck, −bn, −bt, cf, bi,        by, −ca, −bg, −cd, by, bl, ci, −bq}    -   {di, −eq, −cv, −dv, ed, cn, ei, −dq, −da, −ev, dd, dn, −el, −cq,        −ea, dy, cs, en, −dl, −df, et, cy, ds, −eg, −cl, −ef, dt, cx,        es, −dg, −dk, eo, ct, dx, −eb, −cp, −ek, d_, dc, −ew, −db, −dp,        ej, co, ec, −dw, −cu, −ep, dj, dh, −er, −cw, −du, ee, cm, eh,        −dr, −cz, −eu, de, dm, −em, −cr, −dz}    -   {ae, −ag, −ad, −af, af, ad, ag, −ae, −ae, ag, ad, af, −af, −ad,        −ag, ae, ae, −ag, −ad, −af, af, ad, ag, −ae, −ae, ag, ad, af,        −af, −ad, −ag, ae, ae, −ag, −ad, −af, af, ad, ag, −ae, −ae, ag,        ad, af, −af, −ad, −ag, ae, ae, −ag, −ad, −af, af, ad, ag, −ae,        −ae, ag, ad, af, −af, −ad, −ag, ae}    -   {dj, −en, −cq, −ec, du, cy, ev, −db, −dr, ef, cn, ek, −dm, −dg,        eq, ct, dz, −dx, −cv, −es, de, d_, −ei, −cl, −eh, dp, dd, −et,        −cw, −dw, ea, cs, ep, −dh, −dl, el, co, ee, −ds, −da, ew, cz,        dt, −ed, −cp, −em, dk, di, −eo, −cr, −eb, dv, cx, eu, −dc, −dq,        eg, cm, ej, −dn, −df, er, cu, dy}    -   {br, −cf, −bg, −cc, bu, bo, −ci, −bj, −bz, bx, bl, ck, −bm, −bw,        ca, bi, ch, −bp, −bt, cd, bf, ce, −bs, −bq, cg, bh, cb, −by,        −bn, cj, bk, by, −by, −bk, −cj, bn, by, −cb, −bh, −cg, bq, bs,        −ce, −bf, −cd, bt, bp, −ch, −bi, −ca, bw, bm, −ck, −bl, −bx, bz,        bj, ci, −bo, −bu, cc, bg, cf, −br}    -   {dk, −ek, −cl, −ej, dl, dj, −el, −cm, −ei, dm, di, −em, −cn,        −eh, dn, dh, −en, −co, −eg, −d_, dg, −eo, −cp, −ef, dp, df, −ep,        −cq, −ee, dq, de, −eq, −cr, −ed, dr, dd, −er, −cs, −ec, ds, dc,        −es, −ct, −eb, dt, db, −et, −cu, −ea, du, da, −eu, −cv, −dz, dv,        cz, −ev, −cw, −dy, dw, cy, −ew, −cx, −dx}    -   {av, −bb, −ap, −bc, au, aw, −ba, −aq, −bd, at, ax, −az, −ar,        −be, as, ay, −ay, −as, be, ar, az, −ax, −at, bd, aq, ba, −aw,        −au, bc, ap, bb, −av, −av, bb, ap, bc, −au, −aw, ba, aq, bd,        −at, −ax, az, ar, be, −as, −ay, ay, as, −be, −ar, −az, ax, at,        −bd, −aq, −ba, aw, au, −bc, −ap, −bb, av}    -   {dl, −eh, −cp, −eq, dc, du, −dy, −cy, eu, ct, ed, −dp, −dh, el,        cl, em, −dg, −dq, ec, cu, ev, −cx, −dz, dt, dd, −ep, −co, −ei,        dk, dm, −eg, −cq, −er, db, dv, −dx, −cz, et, cs, ee, −d_, −di,        ek, cm, en, −df, −dr, eb, cv, ew, −cw, −ea, ds, de, −eo, −cn,        −ej, dj, dn, −ef, −cr, −es, da, dw}    -   {bs, −cc, −bi, −cj, bl, bz, −by, −bp, cf, bf, cg, −bo, −bw, by,        bm, −ci, −bh, −cd, br, bt, −cb, −bj, −ck, bk, ca, −bu, −bq, ce,        bg, ch, −bn, −bx, bx, bn, −ch, −bg, −ce, bq, bu, −ca, −bk, ck,        bj, cb, −bt, −br, cd, bh, ci, −bm, −by, bw, bo, −cg, −bf, −cf,        bp, by, −bz, −bl, cj, bi, cc, −bs}    -   {dm, −ee, −cu, ew, ct, ef, −dl, −dn, ed, cv, −ev, −cs, −eg, dk,        −d_, −ec, −cw, eu, cr, eh, −dj, −dp, eb, cx, −et, −cq, −ei, di,        dq, −ea, −cy, es, cp, ej, −dh, −dr, dz, cz, −er, −co, −ek, dg,        ds, −dy, −da, eq, cn, el, −df, −dt, dx, db, −ep, −cm, −em, de,        du, −dw, −dc, eo, cl, en, −dd, −dv}    -   {ak, −am, −ai, ao, ah, an, −aj, −al, al, aj, −an, −ah, −ao, ai,        am, −ak, −ak, am, ai, −ao, −ah, −an, aj, al, −al, −aj, an, ah,        ao, −ai, −am, ak, ak, −am, −ai, ao, ah, an, −aj, −al, al, aj,        −an, −ah, −ao, ai, am, −ak, −ak, am, ai, −ao, −ah, −an, aj, al,        −al, −aj, an, ah, ao, −ai, −am, ak}    -   {dn, −eb, −cz, ep, cl, eq, −cy, −ec, dm, d_, −ea, −da, eo, cm,        er, −cx, −ed, dl, dp, −dz, −db, en, cn, es, −cw, −ee, dk, dq,        −dy, −dc, em, co, et, −cv, −ef, dj, dr, −dx, −dd, el, cp, eu,        −cu, −eg, di, ds, −dw, −de, ek, cq, ev, −ct, −eh, dh, dt, −dv,        −df, ej, cr, ew, −cs, −ei, dg, du}    -   {bt, −bz, −bn, cf, bh, ck, −bi, −ce, bo, by, −bu, −bs, ca, bm,        −cg, −bg, −cj, bj, cd, −bp, −bx, by, br, −cb, −bl, ch, bf, ci,        −bk, −cc, bq, bw, −bw, −bq, cc, bk, −ci, −bf, −ch, bl, cb, −br,        −by, bx, bp, −cd, −bj, cj, bg, cg, −bm, −ca, bs, bu, −by, −bo,        ce, bi, −ck, −bh, −cf, bn, bz, −bt}    -   {d_, −dy, −de, ei, cu, −es, −cl, −er, cv, eh, −df, −dx, dp, dn,        −dz, −dd, ej, ct, −et, −cm, −eq, cw, eg, −dg, −dw, dq, dm, −ea,        −dc, ek, cs, −eu, −cn, −ep, cx, ef, −dh, −dv, dr, dl, −eb, −db,        el, cr, −ev, −co, −eo, cy, ee, −di, −du, ds, dk, −ec, −da, em,        cq, −ew, −cp, −en, cz, ed, −dj, −dt}    -   {aw, −ay, −au, ba, as, −bc, −aq, be, ap, bd, −ar, −bb, at, az,        −av, −ax, ax, av, −az, −at, bb, ar, −bd, −ap, −be, aq, bc, −as,        −ba, au, ay, −aw, −aw, ay, au, −ba, −as, bc, aq, −be, −ap, −bd,        ar, bb, −at, −az, av, ax, −ax, −av, az, at, −bb, −ar, bd, ap,        be, −aq, −bc, as, ba, −au, −ay, aw}    -   {dp, −dv, −dj, eb, dd, −eh, −cx, en, cr, −et, −cl, −eu, cq, eo,        −cw, −ei, dc, ec, −di, −dw, −d_, dq, −du, −dk, ea, de, −eg, −cy,        em, cs, −es, −cm, −ev, cp, ep, −cv, −ej, db, ed, −dh, −dx, dn,        dr, −dt, −dl, dz, df, −ef, −cz, el, ct, −er, −cn, −ew, co, eq,        −cu, −ek, da, ee, −dg, −dy, dm, ds}    -   {bu, −bw, −bs, by, bq, −ca, −bo, cc, bm, −ce, −bk, cg, bi, −ci,        −bg, ck, bf, cj, −bh, −ch, bj, cf, −bl, −cd, bn, cb, −bp, −bz,        br, bx, −bt, −by, by, bt, −bx, −br, bz, bp, −cb, −bn, cd, bl,        −cf, −bj, ch, bh, −cj, −bf, −ck, bg, ci, −bi, −cg, bk, ce, −bm,        −cc, bo, ca, −bq, −by, bs, bw, −bu}    -   {dq, −ds, −d_, du, dm, −dw, −dk, dy, di, −ea, −dg, ec, de, −ee,        −dc, eg, da, −ei, −cy, ek, cw, −em, −cu, eo, cs, −eq, −cq, es,        co, −eu, −cm, ew, cl, ev, −cn, −et, cp, er, −cr, −ep, ct, en,        −cv, −el, cx, ej, −cz, −eh, db, ef, −dd, −ed, df, eb, −dh, −dz,        dj, dx, −dl, −dv, dn, dt, −dp, −dr}    -   {aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa,        −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa,        aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa,        −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa,        aa, −aa, −aa, aa, aa, −aa, −aa, aa}    -   {dr, −dp, −dt, dn, dv, −dl, −dx, dj, dz, −dh, −eb, df, ed, −dd,        −ef, db, eh, −cz, −ej, cx, el, −cv, −en, ct, ep, −cr, −er, cp,        et, −cn, −ev, cl, −ew, −cm, eu, co, −es, −cq, eq, cs, −eo, −cu,        em, cw, −ek, −cy, ei, da, −eg, −dc, ee, de, −ec, −dg, ea, di,        −dy, −dk, dw, dm, −du, −d_, ds, dq}    -   {bv, −bt, −bx, br, bz, −bp, −cb, bn, cd, −bl, −cf, bj, ch, −bh,        −cj, bf, −ck, −bg, ci, bi, −cg, −bk, ce, bm, −cc, −bo, ca, bq,        −by, −bs, bw, bu, −bu, −bw, bs, by, −bq, −ca, bo, cc, −bm, −ce,        bk, cg, −bi, −ci, bg, ck, −bf, cj, bh, −ch, −bj, cf, bl, −cd,        −bn, cb, bp, −bz, −br, bx, bt, −bv}    -   {ds, −dm, −dy, dg, ee, −da, −ek, cu, eq, −co, −ew, cn, −er, −ct,        el, cz, −ef, −df, dz, dl, −dt, −dr, dn, dx, −dh, −ed, db, ej,        −cv, −ep, cp, ev, −cm, es, Cs, −em, eg, de, −ea, −dk, du, dq,        −d_, −dw, di, ec, −dc, −ei, cw, eo, −cq, −eu, cl, −et, −cr, en,        cx, −eh, −dd, eb, dj, −dv, −dp}    -   {ax, −av, −az, at, bb, −ar, −bd, ap, −be, −aq, bc, as, −ba, −au,        ay, aw, −aw, −ay, au, ba, −as, −bc, aq, be, −ap, bd, ar, −bb,        −at, az, av, −ax, −ax, av, az, −at, −bb, ar, bd, −ap, be, aq,        −bc, −as, ba, au, −ay, −aw, aw, ay, −au, −ba, as, bc, −aq, −be,        ap, −bd, −ar, bb, at, −az, −av, ax}    -   {dt, −dj, −ed, cz, en, −cp, ew, cq, −em, −da, ec, dk, −ds, −du,        di, ee, −cy, −eo, co, −ev, −cr, el, db, −eb, −dl, dr, dv, −dh,        −ef, cx, ep, −cn, eu, cs, −ek, −dc, ea, dm, −dq, −dw, dg, eg,        −cw, −eq, cm, −et, −ct, ej, dd, −dz, −dn, dp, dx, −df, −eh, cv,        er, −cl, es, cu, −ei, −de, dy, d_}    -   {bw, −bq, −cc, bk, ci, −bf, ch, bl, −cb, −br, by, bx, −bp, −cd,        bj, cj, −bg, cg, bm, −ca, −bs, bu, by, −bo, −ce, bi, ck, −bh,        cf, bn, −bz, −bt, bt, bz, −bn, −cf, bh, −ck, −bi, ce, bo, −by,        −bu, bs, ca, −bm, −cg, bg, −cj, −bj, cd, bp, −bx, −by, br, cb,        −bl, −ch, bf, −ci, −bk, cc, bq, −bw}    -   {du, −dg, −ei, cs, ew, −cr, ej, df, −dv, −dt, dh, eh, −ct, −ev,        cq, −ek, −de, dw, ds, −di, −eg, cu, eu, −cp, el, dd, −dx, −dr,        dj, ef, −cv, −et, co, −em, −dc, dy, dq, −dk, −ee, cw, es, −cn,        en, db, −dz, −dp, dl, ed, −cx, −er, cm, −eo, −da, ea, −dm, −ec,        cy, eq, −cl, ep, cz, −eb, −dn}    -   {al, −aj, −an, ah, −ao, −ai, am, ak, −ak, −am, ai, ao, −ah, an,        aj, −al, −al, aj, an, −ah, ao, ai, −am, −ak, ak, am, −ai, −ao,        ah, −an, −aj, al, al, −aj, −an, ah, −ao, −ai, am, ak, −ak, −am,        ai, ao, −ah, an, aj, −al, −al, aj, an, −ah, ao, ai, −am, −ak,        ak, am, −ai, −ao, ah, −an, −aj, al}    -   {dv, −dd, −en, cl, −eo, −dc, dw, du, −de, −em, cm, −ep, −db, dx,        dt, −df, −el, cn, −eq, −da, dy, ds, −dg, −ek, co, −er, −cz, dz,        dr, −dh, −ej, cp, −es, −cy, ea, dq, −di, −ei, cq, −et, −cx, eb,        dp, −dj, −eh, cr, −eu, −cw, ec, d_, −dk, −eg, cs, −ev, −cv, ed,        dn, −dl, −ef, ct, −ew, −cu, ee, dm}    -   {bx, −bn, −ch, bg, −ce, −bq, bu, ca, −bk, −ck, bj, −cb, −bt, br,        cd, −bh, ci, bm, −by, −bw, bo, cg, −bf, cf, bp, −by, −bz, bl,        cj, −bi, cc, bs, −bs, −cc, bi, −cj, −bl, bz, by, −bp, −cf, bf,        −cg, −bo, bw, by, −bm, −ci, bh, −cd, −br, bt, cb, −bj, ck, bk,        −ca, −bu, bq, ce, −bg, ch, bn, −bx}    -   {dw, −da, −es, cr, −ef, −dn, dj, ej, −cn, eo, de, −ds, −ea, cw,        ew, −cv, eb, dr, −df, −en, cm, −ek, −di, ee, −cs, et, cz, −dx,        −dv, db, er, −cq, eg, dm, −dk, −ei, co, −ep, −dd, dt, dz, −cx,        −ev, cu, −ec, −dq, dg, em, −cl, el, dh, −dp, −ed, ct, −eu, −cy,        dy, du, −dc, −eq, cp, −eh, −di}    -   {ay, −as, −be, ar, −az, −ax, at, bd, −aq, ba, aw, −au, −bc, ap,        −bb, −av, av, bb, −ap, bc, au, −aw, −ba, aq, −bd, −at, ax, az,        −ar, be, as, −ay, −ay, as, be, −ar, az, ax, −at, −bd, aq, −ba,        −aw, au, bc, −ap, bb, av, −av, −bb, ap, −bc, −au, aw, ba, −aq,        bd, at, −ax, −az, ar, −be, −as, ay dx, −cx, ew, cy, −dw, −dy,        cw, −ev, −cz, dv, dz, −cv, eu, da, −du, −ea, cu, −et, −db, dt,        eb, −ct, es, dc, −ds, −ec, cs, −er, −dd, dr, ed, −cr, eq, de,        −dq, −ee, cq, −ep, −df, dp, ef, −cp, eo, dg, −eg, co, −en, −dh,        dn, eh, −cn, em, di, −dm, −ei, cm, −el, −dj, dl, ej, −cl, ek,        dk}    -   {by, −bk, cj, bn, −by, −cb, bh, −cg, −bq, bs, ce, −bf, cd, bt,        −bp, −ch, bi, −ca, −bw, bm, ck, −bl, bx, bz, −bj, ci, bo, −bu,        −cc, bg, −cf, −br, br, cf, −bg, cc, bu, −bo, −ci, bj, −bz, −bx,        bl, −ck, −bm, bw, ca, −bi, ch, bp, −bt, −cd, bf, −ce, −bs, bq,        cg, −bh, cb, by, −bn, −cj, bk, −by}    -   {dy, −cu, er, df, −dn, −ej, cm, −eg, −dq, dc, eu, −cx, dv, eb,        −cr, eo, di, −dk, −em, cp, −ed, −dt, cz, −ew, −da, ds, ee, −co,        el, dl, −dh, −ep, cs, −ea, −dw, cw, −et, −dd, dp, eh, −cl, ei,        −de, −es, cv, −dx, −dz, ct, −eq, −dg, dm, ek, −cn, ef, dr, −db,        −ev, cy, −du, −ec, cq, −en, −dj}    -   {af, −ad, ag, ae, −ae, −ag, ad, −af, −af, ad, −ag, −ae, ae, ag,        −ad, af, af, −ad, ag, ae, −ae, −ag, ad, −af, −af, ad, −ag, −ae,        ae, ag, −ad, af, af, −ad, ag, ae, −ae, −ag, ad, −af, −af, ad,        −ag, −ae, ae, ag, −ad, af, af, −ad, ag, ae, −ae, −ag, ad, −af,        −af, ad, −ag, −ae, ae, ag, −ad, of}    -   {dz, −cr, em, dm, −de, −eu, cz, −dr, −eh, cm, −ee, −du, cw, −er,        −dh, dj, ep, −cu, dw, ec, −co, ej, dp, −db, ew, dc, −ek, cp,        −eb, −dx, ct, −eo, −dk, dg, es, −cx, dt, ef, −cl, eg, ds, −cy,        et, df, −dl, −en, cs, −dy, −ea, cq, −el, −dn, dd, ev, −da, dq,        ei, −cn, ed, dv, −cv, eq, di}    -   {bz, −bh, ce, bu, −bm, cj, bp, −br, −ch, bk, −bw, −cc, bf, −cb,        −bx, bj, −cg, −bs, bo, ck, −bn, bt, cf, −bi, by, ca, −bg, cd,        by, −bl, ci, bq, −bq, −ci, bl, −by, −cd, bg, −ca, −by, bi, −cf,        −bt, bn, −ck, −bo, bs, cg, −bj, bx, cb, −bf, cc, bw, −bk, ch,        br, −bp, −cj, bm, −bu, −ce, bh, −bz}    -   {ea, −co, eh, dt, −cv, eo, dm, −dc, ev, df, −dj, −er, cy, −dq,        −ek, cr, −dx, −ed, cl, −ee, −dw, cs, −el, −dp, cz, −es, −di, dg,        eu, −db, dn, en, −cu, du, eg, −cn, eb, dz, −cp, ei, ds, −cw, ep,        dl, −dd, ew, de, −dk, −eq, cx, −dr, −ej, cq, −dy, −ec, cm, −ef,        −dv, ct, −em, da, −et, −dh}    -   {az, −ap, ba, ay, −aq, bb, ax, −ar, bc, aw, −as, bd, av, −at,        be, au, −au, −be, at, −av, −bd, as, −aw, −bc, ar, −ax, −bb, aq,        −ay, −ba, ap, −az, −az, ap, −ba, −ay, aq, −bb, −ax, ar, −bc,        −aw, as, −bd, −av, at, −be, −au, au, be, −at, av, bd, −as, aw,        bc, −ar, ax, bb, −aq, ay, ba, −ap, az}    -   {eb, −cl, ec, ea, −cm, ed, dz, −cn, ee, dy, −co, ef, dx, −cp,        eg, dw, −cq, eh, dv, −cr, ei, du, −cs, ej, dt, −ct, ek, ds, −cu,        el, dr, −cv, em, dq, −cw, en, dp, −cx, eo, −cy, ep, dn, −cz, eq,        dm, −da, er, dl, −db, es, dk, −dc, et, dj, −dd, eu, di, −de, ev,        dh, −df, ew, dg}    -   {ca, −bf, bz, cb, −bg, by, cc, −bh, bx, cd, −bi, bw, ce, −bj,        by, cf, −bk, bu, cg, −bl, bt, ch, −bm, bs, ci, −bn, br, cj, −bo,        bq, ck, −bp, bp, −ck, −bq, bo, −cj, −br, bn, −ci, −bs, bm, −ch,        −bt, bl, −cg, −bu, bk, −cf, −by, bj, −ce, −bw, bi, −cd, −bx, bh,        −cc, −by, bg, −cb, −bz, bf, −ca}    -   {ec, −cn, dx, eh, −cs, ds, em, −cx, dn, er, −dc, di, ew, −dh,        dd, −es, −dm, cy, −en, −dr, ct, −ei, −dw, co, −ed, −eb, cm, −dy,        −eg, cr, −dt, −el, cw, −eq, db, −dj, −ev, dg, −de, et, dl, −cz,        eo, dq, −cu, ej, dv, −cp, ee, ea, −cl, dz, ef, −cq, du, ek, −cv,        dp, ep, −da, dk, eu, −df}    -   {am, −ah, al, an, −ai, ak, ao, −aj, aj, −ao, −ak, ai, −an, −al,        ah, −am, −am, ah, −al, −an, ai, −ak, −ao, aj, −aj, ao, ak, −ai,        an, al, −ah, am, am, −ah, al, an, −ai, ak, ao, −aj, aj, −ao,        −ak, ai, −an, −al, ah, −am, −am, ah, −al, −an, ai, −ak, −ao, aj,        −aj, ao, ak, −ai, an, al, −ah, am}    -   {ed, −cq, ds, eo, −db, dh, −eu, −dm, cw, −ej, −dx, cl, −dy, −ei,        cv, −dn, −et, dg, −dc, ep, dr, −cr, ee, ec, −cp, dt, en, −da,        di, −ev, −dl, cx, −ek, −dw, cm, −dz, −eh, cu, −d_, −es, df, −dd,        eq, dq, −cs, ef, eb, −co, du, em, −cz, dj, −ew, −dk, cy, −el,        −dv, cn, −ea, −eg, ct, −dp, −er, de}    -   {cb, −bi, bu, ci, −bp, bn, −cg, −bw, bg, −bz, −cd, bk, −bs, −ck,        br, −bl, ce, by, −bf, bx, cf, −bm, bq, −cj, −bt, bj, −cc, −ca,        bh, −by, −ch, bo, −bo, ch, by, −bh, ca, cc, −bj, bt, cj, −bq,        bm, −cf, −bx, bf, −by, −ce, bl, −br, ck, bs, −bk, cd, bz, −bg,        bw, cg, −bn, bp, −ci, −bu, bi, −cb}    -   {ee, −ct, dn, ev, −dk, cw, −eh, −eb, cq, −dq, −es, dh, −cz, ek,        dy, −cn, dt, ep, −de, dc, −en, −dv, cl, −dw, −em, db, −df, eq,        ds, −co, dz, ej, −cy, di, −et, −dp, cr, −ec, −eg, cv, −dl, ew,        dm, −cu, ef, ed, −cs, d_, eu, −dj, cx, −ei, −ea, cp, −dr, −er,        dg, −da, el, dx, −cm, du, eo, −dd}    -   {ba, −ar, av, −be, −aw, aq, −az, −bb, as, −au, bd, ax, −ap, ay,        bc, −at, at, −bc, −ay, ap, −ax, −bd, au, −as, bb, az, −aq, aw,        be, −av, ar, −ba, −ba, ar, −av, be, aw, −aq, az, bb, −as, au,        −bd, −ax, ap, −ay, −bc, at, −at, bc, ay, −ap, ax, bd, −au, as,        −bb, −az, aq, −aw, −be, av, −ar, ba}    -   {ef, −cw, di, −er, −dt, cl, −du, −eq, dh, −cx, eg, ee, −cv, dj,        −es, −ds, cm, −dv, −ep, dg, −cy, eh, ed, −cu, dk, −et, −dr, cn,        −dw, −eo, df, −cz, ei, ec, −ct, dl, −eu, −dq, co, −dx, −en, de,        −da, ej, eb, −cs, dm, −ev, −dp, cp, −dy, −em, dd, −db, ek, ea,        −cr, dn, −ew, cq, −dz, −el, dc}    -   {cc, −bl, bp, −cg, −by, bh, −bt, ck, bu, −bg, bx, ch, −bq, bk,        −cb, −cd, bm, −bo, cf, bz, −bi, bs, −cj, −by, bf, −bw, −ci, br,        −bj, ca, ce, −bn, bn, −ce, −ca, bj, −br, ci, bw, −bf, by, cj,        −bs, bi, −bz, −cf, bo, −bm, cd, cb, −bk, bq, −ch, −bx, bg, −bu,        −ck, bt, −bh, by, cg, −bp, bl, −cc}    -   {eg, −cz, dd, −ek, −ec, cv, −dh, eo, dy, −cr, dl, −es, −du, cn,        −dp, ew, dq, −cm, dt, et, −dm, cq, −dx, −ep, di, −cu, eb, el,        −de, cy, −ef, −eh, da, −dc, ej, ed, −cw, dg, −en, −dz, cs, −dk,        er, dv, −co, −ev, −dr, cl, −ds, −eu, dn, −cp, dw, eq, −dj, ct,        −ea, −em, df, −cx, ee, ei, −db}    -   {ac, −ab, ab, −ac, −ac, ab, −ab, ac, ac, −ab, ab, −ac, −ac, ab,        −ab, ac, ac, −ab, ab, −ac, −ac, ab, −ab, ac, ac, −ab, ab, −ac,        −ac, ab, −ab, ac, ac, −ab, ab, −ac, −ac, ab, −ab, ac, ac, −ab,        ab, −ac, −ac, ab, −ab, ac, ac, −ab, ab, −ac, −ac, ab, −ab, ac,        ac, −ab, ab, −ac, −ac, ab, −ab, ac}    -   {eh, −dc, cy, −ed, −el, dg, −cu, dz, ep, −dk, cq, −dv, −et, −cm,        dr, −ew, −ds, cn, −dn, es, dw, −cr, dj, −eo, −ea, cv, −df, ek,        ee, −cz, db, −eg, −ei, dd, −cx, ec, em, −dh, ct, −dy, −eq, dl,        −cp, du, eu, −dp, cl, −dq, ev, dt, −co, dm, −er, −dx, cs, −di,        en, eb, −cw, de, −ej, −ef, da}    -   {cd, −bo, bk, −bz, −ch, bs, −bg, by, −ck, −bw, bh, −br, cg, ca,        −bl, bn, −cc, −ce, bp, −bj, by, ci, −bt, bf, −bu, cj, bx, −bi,        bq, −cf, −cb, bm, −bm, cb, cf, −bq, bi, −bx, −cj, bu, −bf, bt,        −ci, −by, bj, −bp, ce, cc, −bn, bl, −ca, −cg, br, −bh, bw, ck,        −by, bg, −bs, ch, bz, −bk, bo, −cd}    -   {ei, −df, ct, −dw, −eu, dr, −co, dk, −en, −ed, da, eb, ep, −dm,        cm, −dp, es, dy, −cv, dd, −eg, −ek, dh, −cr, du, ew, −dt, cq,        −di, el, ef, −dc, cw, −dz, −er, −cl, dn, −eq, −ea, cx, −db, ee,        em, −dj, cp, −ds, ev, dv, −cs, dg, −ej, −eh, de, −cu, dx, et,        −dq, cn, −dl, eo, ec, −cz}    -   {bb, −au, aq, −ax, be, ay, −ar, at, −ba, −bc, av, −ap, aw, −bd,        −az, as, −as, az, bd, −aw, ap, −av, bc, ba, −at, ar, −ay, −be,        ax, −aq, au, −bb, −bb, au, −aq, ax, −be, −ay, ar, −at, ba, bc,        −av, ap, −aw, bd, az, −as, as, −az, −bd, aw, −ap, av, −bc, −ba,        at, −ar, ay, be, −ax, aq, −au, bb}    -   {ej, −di, co, −dp, eq, ec, −db, cv, −dw, −ew, dv, −cu, dc, −ed,        −ep, −cn, dj, −ek, −ei, dh, −cp, dq, −er, −eb, da, −cw, dx, ev,        −du, ct, −dd, ee, eo, −dn, cm, −dk, el, eh, −dg, cq, −dr, es,        ea, −cz, cx, −dy, −eu, dt, −cs, de, −ef, −en, dm, −cl, dl, −em,        −eg, df, −cr, ds, −et, −dz, cy}    -   {ce, −br, bf, −bs, cf, cd, −bq, bg, −bt, cg, cc, −bp, bh, −bu,        ch, cb, −bo, bi, −by, ci, ca, −bn, bj, −bw, cj, bz, −bm, bk,        −bx, ck, by, −bl, bl, −by, −ck, bx, −bk, bm, −bz, −cj, bw, −bj,        bn, −ca, −ci, by, −bi, bo, −cb, −ch, bu, −bh, bp, −cc, −cg, bt,        −bg, bq, −cd, −cf, bs, −bf, br, −ce}    -   {ek, −dl, cm, −di, eh, en, cp, −df, ee, eq, −dr, cs, −dc, eb,        et, −du, cv, −cz, dy, ew, −dx, cy, −cw, dv, −eu, −ea, db, −ct,        ds, −er, −ed, de, −cq, dp, −eo, −eg, dh, −cn, dm, −el, −ej, dk,        −cl, dj, −ei, −em, dn, −co, dg, −ef, −ep, dq, −cr, dd, −ec, −es,        dt, −cu, da, −dz, −ev, dw, −cx}    -   {an, −ak, ah, −aj, am, ao, −al, ai, −ai, al, −ao, −am, aj, −ah,        ak, −an, −an, ak, −ah, aj, −am, −ao, al, −ai, ai, −al, ao, am,        −aj, ah, −ak, an, an, −ak, ah, −aj, am, ao, −al, ai, −ai, al,        −ao, −am, aj, −ah, ak, −an, −an, ak, −ah, aj, −am, −ao, al, −ai,        ai, −al, ao, am, −aj, ah, −ak, an}    -   {el, cr, −db, dy, −ev, −eb, de, −co, dl, −ei, −eo, dr, −cu, cy,        −dv, es, ee, −dh, cl, −di, ef, er, −du, cx, −cv, ds, −ep, −eh,        dk, −cn, df, −ec, −eu, dx, −da, cs, −dp, em, ek, −dn, cq, −dc,        dz, −ew, −ea, dd, −cp, dm, −ej, −en, dq, −ct, cz, −dw, et, ed,        −dg, cm, −dj, eg, eq, −dt, cw}    -   {cf, −bu, bj, −bl, bw, −ch, −cd, bs, −bh, bn, −by, cj, cb, −bq,        bf, −bp, ca, ck, −bz, bo, −bg, br, −cc, −ci, bx, −bm, bi, −bt,        ce, cg, −by, bk, −bk, by, −cg, −ce, bt, −bi, bm, −bx, ci, cc,        −br, bg, −bo, bz, −ck, −ca, bp, −bf, bq, −cb, −cj, by, −bn, bh,        −bs, cd, ch, −bw, bl, −bj, bu, −cf}    -   {em, −dr, cw, −cu, dp, −ek, −eo, dt, cs, −dn, ei, eq, −dv, da,        −cq, dl, −eg, −es, dx, −dc, co, −dj, ee, eu, −dz, de, −cm, dh,        −ec, −ew, eb, −dg, cl, −df, ea, −ev, −ed, di, −cn, dd, −dy, et,        ef, −dk, cp, −db, dw, −er, −eh, dm, −cr, cz, −du, ep, ej, ct,        −cx, ds, −en, −el, dq, −cv}    -   {bc, −ax, as, −aq, av, −ba, −be, az, −au, ap, −at, ay, −bd, −bb,        aw, −ar, ar, −aw, bb, bd, −ay, at, −ap, au, −az, be, ba, −av,        aq, −as, ax, −bc, −bc, ax, −as, aq, −av, ba, be, −az, au, −ap,        at, −ay, bd, bb, −aw, ar, −ar, aw, −bb, −bd, ay, −at, ap, −au,        az, −be, −ba, av, −aq, as, −ax, bc}    -   {en, −du, db, −cn, dg, −dz, es, ei, −dp, cw, −cs, dl, −ee, −ew,        ed, −dk, cr, −cx, dq, −ej, −er, dy, −df, cm, −dc, dv, −eo, −em,        dt, −da, co, −dh, ea, −et, −eh, −cv, ct, −dm, ef, ev, −ec, dj,        −cq, cy, −dr, ek, eq, −dx, de, −cl, dd, −dw, ep, el, −ds, cz,        −cp, di, −eb, eu, eg, −dn, cu}    -   {cg, −bx, bo, −bf, bn, −bw, cf, ch, −by, bp, −bg, bm, −by, ce,        ci, −bz, bq, −bh, bl, −bu, cd, cj, −ca, br, −bi, bk, −bt, cc,        ck, −cb, bs, −bj, bj, −bs, cb, −ck, −cc, bt, −bk, bi, −br, ca,        −cj, −cd, bu, −bl, bh, −bq, bz, −ci, −ce, by, −bm, bg, −bp, by,        −ch, −cf, bw, −bn, bf, −bo, bx, −cg}    -   {eo, −dx, dg, −cp, cx, ef, −ew, −eg, dp, −cy, co, −df, dw, −en,        −ep, dy, −dh, cq, −cw, dn, −ee, ev, eh, −dq, cz, −cn, de, −dv,        em, eq, −dz, di, −cr, cv, −dm, ed, −eu, −ei, dr, −da, cm, −dd,        du, −el, −er, ea, −dj, cs, −cu, dl, −ec, et, ej, −ds, db, −cl,        dc, −dt, ek, es, −eb, dk, −ct}    -   {ag, −af, ae, −ad, ad, −ae, af, −ag, −ag, af, −ae, ad, −ad, ae,        −af, ag, ag, −af, ae, −ad, ad, −ae, af, −ag, −ag, af, −ae, ad,        −ad, ae, −af, ag, ag, −af, ae, −ad, ad, −ae, af, −ag, −ag, af,        −ae, ad, −ad, ae, −af, ag, ag, −af, ae, −ad, ad, −ae, af, −ag,        −ag, af, −ae, ad, −ad, ae, −af, ag}    -   {ep, −ea, dl, −cw, co, −dd, ds, −eh, ew, ei, −dt, de, −cp, cv,        −dk, dz, −eo, −eq, eb, −dm, cx, −cn, dc, −dr, eg, −ev, −ej, du,        −df, cq, −cu, dj, −dy, en, er, −ec, dn, −cy, cm, −db, dq, −ef,        eu, ek, −dv, dg, −cr, ct, −di, dx, −em, −es, ed, cz, −cl, da,        −dp, ee, −et, −el, dw, −dh, cs}    -   {ch, −ca, bt, −bm, bf, −bl, bs, −bz, cg, ci, −cb, bu, −bn, bg,        −bk, br, −by, cf, cj, −cc, by, −bo, bh, −bj, bq, −bx, ce, ck,        −cd, bw, −bp, bi, −bi, bp, −bw, cd, −ck, −ce, bx, −bq, bj, −bh,        bo, −by, cc, −cj, −cf, by, −br, bk, −bg, bn, −bu, cb, −ci, −cg,        bz, −bs, bl, −bf, bm, −bt, ca, −ch}    -   {eq, −ed, dq, −dd, cq, −cs, df, −ds, ef, −es, −eo, eb, db, −co,        cu, −dh, du, −eh, eu, em, −dz, dm, −cz, cm, −cw, dj, −dw, ej,        −ew, −ek, dx, −dk, cx, −cl, cy, −dl, dy, −el, −ev, ei, −dv, di,        −cv, cn, −da, dn, −ea, en, et, −eg, dt, −dg, ct, −cp, dc, −dp,        ec, −ep, −er, ee, −dr, de, −cr}    -   {bd, −ba, ax, −au, ar, −ap, as, −av, ay, −bb, be, bc, −az, aw,        −at, aq, −aq, at, −aw, az, −bc, −be, bb, −ay, av, −as, ap, −ar,        au, −ax, ba, −bd, −bd, ba, −ax, au, −ar, ap, −as, av, −ay, bb,        −be, −bc, az, −aw, at, −aq, aq, −at, aw, −az, bc, be, −bb, ay,        −av, as, −ap, ar, −au, ax, −ba, bd}    -   {er, −eg, dv, −dk, cz, −co, cs, −dd, −dz, ek, −ev, −en, ec, −dr,        dg, −cv, cl, −cw, dh, −ds, ed, −eo, −eu, ej, −dy, dn, −dc, cr,        −cp, da, −dl, dw, −eh, es, eq, −ef, du, −dj, cy, −cn, ct, −de,        dp, −ea, el, −ew, −em, eb, −dq, df, −cu, cm, −cx, di, −dt, ee,        −ep, −et, ei, −dx, dm, −db, cq}    -   {ci, −cd, by, −bt, bo, −bj, bf, −bk, bp, −bu, bz, −ce, cj, ch,        −cc, bx, −bs, bn, −bi, bg, −bl, bq, −by, ca, −cf, ck, cg, −cb,        bw, −br, bm, −bh, bh, −bm, br, −bw, cb, −cg, −ck, cf, −ca, by,        −bq, bl, −bg, bi, −bn, bs, −bx, cc, −ch, −cj, ce, −bz, bu, −bp,        bk, −bf, bj, −bo, bt, −by, cd, −ci}    -   {es, −ej, ea, −dr, di, −cz, cq, −co, cx, −dg, dp, −dy, eh, −eq,        −eu, el, −ec, dt, −dk, db, −cs, cm, −cv, de, −dn, dw, −ef, eo,        ew, −en, ee, −dv, dm, −dd, cu, −cl, ct, −dc, dl, −du, ed, −em,        ev, ep, −eg, dx, df, −cw, cn, −cr, da, −dj, ds, −eb, ek, −et,        −er, ei, −dz, dq, −dh, cy, −cp}    -   {ao, −an, am, −al, ak, −aj, ai, −ah, ah, −ai, aj, −ak, al, −am,        an, −ao, −ao, an, −am, al, −ak, aj, −ai, ah, −ah, ai, −aj, ak,        −al, am, −an, ao, ao, −an, am, −al, ak, −aj, ai, −ah, ah, −ai,        aj, −ak, al, −am, an, −ao, −ao, an, −am, al, −ak, aj, −ai, ah,        −ah, ai, −aj, ak, −al, am, −an, ao}    -   {et, −em, ef, −dy, dr, −dk, dd, −cw, cp, −cn, cu, −db, di, −dp,        dw, −ed, ek, −er, −ev, eo, −eh, ea, −dt, dm, −df, cy, −cr, cl,        −cs, cz, −dg, dn, −du, eb, −ei, ep, −ew, −eq, ej, −ec, dv, dh,        −da, ct, −cm, cq, −cx, de, −dl, ds, −dz, eg, −en, eu, es, −el,        ee, −dx, dq, −dj, dc, −cv, co}    -   {cj, −cg, cd, −ca, bx, −bu, br, −bo, bl, −bi, bf, −bh, bk, −bn,        bq, −bt, bw, −bz, cc, −cf, ci, ck, −ch, ce, −cb, by, −by, bs,        −bp, bm, −bj, bg, −bg, bj, −bm, bp, −bs, by, −by, cb, −ce, ch,        −ck, −ci, cf, −cc, bz, −bw, bt, −bq, bn, −bk, bh, −bf, bi, −bl,        bo, −br, bu, −bx, ca, −cd, cg, −el}    -   {eu, −ep, ek, −ef, ea, −dv, dq, −dl, dg, −db, cw, −cr, cm, −co,        ct, −cy, dd, −di, dn, −ds, dx, −ec, eh, −em, er, −ew, −es, en,        −ei, ed, −dy, dt, dj, −de, cz, −cu, cp, −cl, cq, −cv, da, −df,        dk, −dp, du, −dz, ee, −ej, eo, −et, −ev, eq, −el, eg, −eb, dw,        −dr, dm, −dh, dc, −cx, cs, −cn}    -   {be, −bd, bc, −bb, ba, −az, ay, −ax, aw, −av, au, −at, as, −ar,        aq, −ap, ap, −aq, ar, −as, at, −au, av, −aw, ax, −ay, az, −ba,        bb, −bc, bd, −be, −be, bd, −bc, bb, −ba, az, −ay, ax, −aw, av,        −au, at, −as, ar, −aq, ap, −ap, aq, −ar, as, −at, au, −av, aw,        −ax, ay, −az, ba, −bb, bc, −bd, be}    -   {ev, −es, ep, −em, ej, −eg, ed, −ea, dx, −du, dr, dl, −di, df,        −dc, cz, −cw, ct, −cq, cn, −cl, co, −cr, cu, −cx, da, −dd, dg,        −dj, dm, −dp, ds, −dv, dy, −eb, ee, −eh, ek, −en, eq, −et, ew,        eu, −er, eo, −el, ei, −ef, ec, −dz, dw, −dt, dq, −dn, dk, −dh,        de, −db, cy, −cv, cs, −cp, cm}    -   {ck, −cj, ci, −ch, cg, −cf, ce, −cd, cc, −cb, ca, −bz, by, −bx,        bw, −by, bu, −bt, bs, −br, bq, −bp, bo, −bn, bm, −bl, bk, −bj,        bi, −bh, bg, −bf, bf, −bg, bh, −bi, bj, −bk, bl, −bm, bn, −bo,        bp, −bq, br, −bs, bt, −bu, by, −bw, bx, −by, bz, −ca, cb, −cc,        cd, −ce, cf, −cg, ch, −ci, cj, −ck}    -   {ew, −ev, eu, −et, es, −er, eq, −ep, eo, −en, em, −el, ek, −ej,        ei, −eh, eg, −ef, ee, −ed, ec, −eb, ea, −dz, dy, −dx, dw, −dv,        du, −dt, ds, −dr, dq, −dp, −dn, dm, −dl, dk, −dj, di, −dh, dg,        −df, de, −dd, dc, −db, da, −cz, cy, −cx, cw, −cv, cu, −ct, cs,        −cr, cq, −cp, co, −cn, cm, −cl}.

In addition to DCT-2 and 4×4 DST-7, which have been employed in HEVC, anAdaptive Multiple Transform (AMT, or as known as Enhanced MultipleTransform (EMT)) scheme has been proposed for residual coding for bothinter and intra coded blocks. The AMT may use multiple selectedtransforms from the DCT/DST families other than the current transformsin HEVC. The newly introduced transform matrices may include DST-7,DCT-8, DST-1 and DCT-5. Table 1 below shows exemplary basis functions ofa selected DST/DCT.

TABLE 1 Transform basis functions of DCT-II/V/VIII and DST-I/VII forN-point input Transform Type Basis function T_(i)(j), i, j = 0, 1, . . ., N − 1 DCT-2${T_{i}(j)} = {\omega_{0} \cdot \sqrt{\frac{2}{N}} \cdot {\cos\left( \frac{\pi \cdot i \cdot \left( {{2j} + 1} \right)}{2N} \right)}}$${{where}\mspace{14mu}\omega_{0}} = \left\{ \begin{matrix}\sqrt{\frac{2}{N}} & {i = 0} \\1 & {i \neq 0}\end{matrix} \right.$ DCT-5${{T_{i}(j)} = {\omega_{0} \cdot \omega_{1} \cdot \sqrt{\frac{2}{{2N} - 1}} \cdot {\cos\left( \frac{2{\pi \cdot i \cdot j}}{{2N} - 1} \right)}}},$${{where}\mspace{14mu}\omega_{0}} = \left\{ {\begin{matrix}\sqrt{\frac{2}{N}} & {i = 0} \\1 & {i \neq 0}\end{matrix},{\omega_{1} = \left\{ \begin{matrix}\sqrt{\frac{2}{N}} & {j = 0} \\1 & {j \neq 0}\end{matrix} \right.}} \right.$ DCT-8${T_{i}(j)} = {\sqrt{\frac{4}{{2N} + 1}} \cdot {\cos\left( \frac{\pi \cdot \left( {{2i} + 1} \right) \cdot \left( {{2j} + 1} \right)}{{4N} + 2} \right)}}$DST-1${T_{i}(j)} = {\sqrt{\frac{2}{N + 1}} \cdot {\sin\left( \frac{\pi \cdot \left( {i + 1} \right) \cdot \left( {j + 1} \right)}{N + 1} \right)}}$DST-7${T_{i}(j)} = {\sqrt{\frac{4}{{2N} + 1}} \cdot {\sin\left( \frac{\pi \cdot \left( {{2i} + 1} \right) \cdot \left( {j + 1} \right)}{{2N} + 1} \right)}}$

In order to keep the orthogonality of the transform matrix, thetransform matrices may be quantized more accurately than the transformmatrices in HEVC, with 10-bit representation instead of 8-bit in HEVC.To keep the intermediate values of the transformed coefficients withinthe range of 16-bit, after horizontal and after vertical transform, allthe coefficients may be right shifted by 2 more bits compared to theright shift used in the current HEVC transforms. The AMT may apply tocoding units (CUs) with both width and height smaller than or equal to64, and whether AMT applies or not may be controlled by a CU level flag.When the CU level flag is equal to 0, DCT-2 may be applied in the CU toencode the residue. For luma coding block within an AMT-enabled CU, twoadditional flags may be signalled to identify the horizontal andvertical transform to be used.

For intra residue coding, due to the different residual statistics ofdifferent intra prediction modes, a mode-dependent transform candidateselection process may be used. Three transform subsets may be defined asshown in Table 2 below, and the transform subset may be selected basedon the intra prediction mode, as specified in Table 3 below.

TABLE 2 Three pre-defined transform candidate sets Transform SetTransform Candidates 0 DST-7 DCT-8 1 DST-7, DST-1 2 DST-7, DCT-5

With the subset concept, a transform subset may be first identifiedbased on Table 3 using the intra prediction mode of a CU with a CU-levelAMT flag that is equal to 1. After that, for each of the horizontaltransform and the vertical transform, one of the two transformcandidates in the identified transform subset, according to Table 2, maybe selected based on being explicitly signalled with flags.

TABLE 3 Selected (H)orizontal and (V)ertical transform sets for eachintra prediction mode Intra 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17Mode V 2 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 0 H 2 1 0 1 0 1 0 1 0 1 0 1 0 12 2 2 2 Intra 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 Mode V0 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 H 2 2 2 2 2 1 0 1 0 1 0 1 0 1 0 1 0Intra 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 Mode V 1 0 10 1 0 1 0 1 0 1 2 2 2 2 2 2 2 H 1 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0Intra 53 54 55 56 57 58 59 60 61 62 63 64 65 66 Mode V 2 2 1 0 1 0 1 0 10 1 0 1 0 H 0 0 1 0 1 0 1 0 1 0 1 0 1 0

However, for inter prediction residual, only one transform set, whichconsists of DST-7 and DCT-8, may be used for all inter modes and forboth horizontal and vertical transforms. Among the four additionaltransform types, i.e., DST-7, DCT-8, DST-1 and DCT-5, the mostefficiently used transform types are DST-7 and DCT-8. It is noted thatDCT-8 is essentially flipping DST-7 basis left and right with signchanges, so DCT-8 and DST-7 may basically share the same transformbases.

The transform core, which is a matrix composed by the basis vectors, ofDST-7 can be also be represented as shown below:

4-Point DST-7:

-   -   a, b, c, d    -   c, c, 0, −c    -   d, −a, −c, b    -   b, −d, c, −a

8-Point DST-7:

-   -   a, b, c, d, e, f, g, h    -   c, f, h, e, b, −a, −d, −g    -   e, g, b, −c, −h, −d, a, f    -   g, c, −d, −f, a, h, b, −e    -   h, −a, −g, b, f, −c, −e, d    -   f, −e, −a, g, −d, −b, h, −c    -   d, −h, e, −a, −c, g, −f, b    -   b, −d, f, −h, g, −e, c, −a

16-Point PST-7:

-   -   a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p    -   c, f, i, l, o, o, l, i, f, c, 0, −c, −f, −i, −l, −o    -   e, j, o, m, h, c, −b, −g, −l, −p, −k, −f, −a, d, i, n    -   g, n, l, e, −b, −i, −p, −j, −c, d, k, o, h, a, −f, −m    -   i, o, f, −c, −l, −l, −c, f, o, i, 0, −i, −o, −f, c, l    -   k, k, 0, −k, −k, 0, k, k, 0, −k, −k, 0, k, k, 0, −k    -   m, g, −f, −n, −a, l, h, −e, −o, −b, k, i, −d, −p, −c, j    -   o, c, −l, −f, i, i, −f, −l, c, O, 0, −o, −c, l, f, −i    -   p, −a, −o, b, n, −c, −m, d, l, −e, −k, f, j, −g, −i, h    -   n, −e, −i, j, d, −o, a, m, −f, −h, k, c, −p, b, l, −g    -   l, −i, −c, o, −f, −f, O, −c, −i, l, 0, −l, i, c, −o, f    -   j, −m, c, g, −p, f, d, −n, i, a, −k, l, −b, −h, o, −e    -   h, −p, i, −a, −g, o, −j, b, f, −n, k, −c, −e, m, −l, d    -   f, −l, o, −i, c, c, −i, o, −l, f, 0, −f, l, −o, i, −c    -   d, −h, l, −p, m, −i, e, −a, −c, g, −k, o, −n, j, −f, b    -   b, −d, f, −h, j, −l, n, −p, o, −m, k, −i, g, −e, c, −a

32-Point DST-7:

-   -   a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u,        v, w, x, y, z, A, B, C, D, E, F    -   c, f, i, l, o, r, u, x, A, D, F, C, z, w, t, q, n, k, h, a, b,        −a, −d, −g, −j, −m, −p, −s, −v, −y, −B, −E    -   e, j, o, t, y, D, D, y, t, o, j, e, 0, −a, −j, −o, −t, −y, −D,        −D, −y, −t, −o, −j, −e, 0, e, j, o, t, y, D    -   g, n, u, B, D, w, p, i, b, −a, −l, −s, −z, −F, −y, −r, −k, −d,        c, j, q, x, E, A, t, m, f, −a, −h, −o, −v, −C    -   i, A, C, t, k, b, −g, −p, −y, −E, −v, −m, −d, e, n, w, F, x, o,        f, −c, −l, −u, −D, −z, −q, −h, a, j, s, B    -   k, v, F, u, j, −a, −l, −w, −E, −t, −i, b, m, x, D, s, h, −c, −n,        −y, −C, −r, −g, d, o, z, B, q, f, −a, −p, −A    -   m, z, z, m, 0, −m, −z, −z, −m, 0, m, z, z, m, 0, −m, −z, −z, −m,        0, m, z, z, m, 0, −m, −z, −z, −m, 0, m, z    -   o, D, t, e, −j, −y, −y, −j, e, t, D, o, 0, −o, −D, −t, −a, j, y,        y, j, −e, −t, −D, −o, 0, o, D, t, e, −j, −y    -   q, E, n, −c, −t, −B, −k, f, w, y, h, −i, −z, −v, −e, l, C, s, b,        −o, −F, −p, a, r, D, m, −d, −u, −A, −j, g, x    -   s, A, h, −k, −D, −p, c, v, x, e, −n, −F, −m, f, y, u, b, −q, −C,        −j, i, B, r, −a, −t, −z, −g, l, E, o, −d, −w    -   u, w, b, −s, −y, −d, q, A, f, −o, −C, −h, m, E, j, −k, −F, −l,        i, D, n, −g, −B, −p, a, z, r, −c, −x, −t, a, v    -   w, s, −d, −A, −o, h, E, k, −l, −D, −g, p, z, c, −t, −v, a, x, r,        −e, −B, −n, i, F, j, −m, −C, −f, q, y, b, −u    -   y, o, −j, −D, −e, t, t, −e, −D, −j, o, y, 0, −y, −o, j, D, e,        −t, −t, a, D, j, −o, −y, 0, y, o, −j, −D, −e, t    -   A, k, −p, −v, e, F, f, −u, −q, j, B, a, −z, −l, o, w, −d, −E,        −g, t, r, −i, −C, −b, y, m, −n, −x, c, D, h, −s    -   C, g, −v, −n, o, u, −h, −B, a, D, f, −w, −m, p, t, −i, −A, b, E,        e, −x, −l, q, s, −j, −z, c, F, d, −y, −k, r    -   E, c, −B, −f, y, i, −v, −l, s, o, −p, −r, m, u, −j, −x, g, A,        −d, −D, a, F, b, −C, −e, z, h, −w, −k, t, n, −q    -   F, −a, −E, b, D, −c, −C, d, B, −e, −A, f, z, −g, −y, h, x, −i,        −w, j, v, −k, −u, l, t, −m, −s, n, r, −o, −q, p    -   D, −e, −y, j, t, −o, −o, t, j, −y, −e, D, 0, −D, e, y, −j, −t,        o, o, −t, −j, y, e, −D, 0, D, −e, −y, j, t, −o    -   B, −i, −s, r, j, −A, −a, C, −h, −t, q, k, −z, −b, D, −g, −u, p,        l, −y, −c, E, −f, −v, o, m, −x, −d, F, −e, −w, n    -   z, −m, −m, z, 0, −z, m, m, −z, 0, z, −m, −m, z, 0, −z, m, m, −z,        0, z, −m, −m, z, 0, −z, m, m, −z, 0, z, −m    -   x, −q, −g, E, −j, −n, A, −c, −u, t, d, −B, m, k, −D, f, r, −w,        −a, y, −p, −h, F, −i, −o, z, −b, −v, s, e, −C, l    -   v, −u, −a, w, −t, −b, x, −s, −c, y, −r, −d, z, −q, −e, A, −p,        −f, B, −o, −g, C, −n, −h, D, −m, −i, E, −l, −j, F, −k    -   t, −y, e, o, −D, j, j, −D, o, e, −y, t, 0, −t, y, −e, −o, D, −j,        −j, D, −o, −e, y, −t, 0, t, −y, e, o, −D, j    -   r, −C, k, g, −y, v, −d, −n, F, −o, −c, u, −z, h, j, −B, s, −a,        −q, D, −l, −f, x, −w, e, m, −E, p, b, −t, A, −i    -   p, −F, q, −a, −o, E, −r, b, n, −D, s, −c, −m, C, −t, d, l, −B,        u, −e, −k, A, −v, f, j, −z, w, −q, y, −x, h    -   n, −B, w, −i, −e, a, −F, r, −d, −j, x, −A, m, a, −o, C, −v, h,        f, −t, E, −q, c, k, −y, z, −l, −b, p, −D, u, −g    -   l, −x, C, −q, e, g, −a, E, −v, j, b, −n, z, −A, o, −c, −i, u,        −F, t, −h, −d, p, −B, y, −m, a, k, −w, D, −r, f    -   j, −t, D, −y, o, −e, −e, o, −y, D, −t, j, 0, −j, t, −D, y, −o,        e, e, −o, y, −D, t, −j, 0, j, −t, D, −y, o, −e    -   h, −p, x, −F, y, −q, i, −a, −g, o, −w, E, −z, r, −j, b, f, −n,        v, −D, A, −s, k, −c, −e, m, −u, C, −B, t, −l, d    -   f, −l, r, −x, D, −C, w, −q, k, −e, −a, g, −m, s, −y, E, −B, v,        −p, j, −d, −b, h, −n, t, −z, F, −A, u, −o, i, −c    -   d, −h, l, −p, t, −x, B, −F, C, −y, u, −q, m, −l, e, −a, −c, g,        −k, o, −s, w, −A, E, −D, z, −v, r, −n, j, −f, b    -   b, −d, f, −h, j, −l, n, −p, r, −t, v, −x, z, −B, D, −F, E, −C,        A, −y, w, −u, s, −q, o, −m, k, g, −e, c, −a

64-Point DST-7:

-   -   {aa, ab, ac, ad, ae, af, ag, ah, ai, aj, ak, al, am, an, ao, ap,        aq, ar, as, at, au, av, aw, ax, ay, az, ba, bb, bc, bd, be, bf,        bg, bh, bi, bj, bk, bl, bm, bn, bo, bp, bq, br, bs, bt, bu, by,        bw, bx, by, bz, ca, cb, cc, cd, ce, cf, cg, ch, ci, cj, ck, cl}    -   {ab, ae, ah, ak, an, aq, at, aw, az, bc, bf, bi, bl, bo, br, bu,        bx, ca, cd, cg, cj, cl, ci, cf, cc, bz, bw, bt, bq, bn, bk, bh,        be, bb, ay, av, as, ap, am, aj, ag, ad, aa, −ac, −af, −ai, −al,        −ao, −ar, −au, −ax, −ba, −bd, −bg, −bj, −bm, −bp, −bs, −by, −by,        −cb, −ce, −ch, −ck}    -   {ac, ah, am, ar, aw, bb, bg, bl, bq, by, ca, cf, ck, ci, cd, by,        bt, bo, bj, be, az, au, ap, ak, af, aa, −ae, −aj, −ao, −at, −ay,        −bd, −bi, −bn, −bs, −bx, −cc, −ch, −cl, −cg, −cb, −bw, −br, −bm,        −bh, −bc, −ax, −as, −an, −ai, −ad, ab, ag, al, aq, av, ba, bf,        bk, bp, bu, bz, ce, cj}    -   {ad, ak, ar, ay, bf, bm, bt, ca, ch, cj, cc, by, bo, bh, ba, at,        am, af, −ab, −ai, −ap, −aw, −bd, −bk, −br, −by, −cf, −cl, −ce,        −bx, −bq, −bj, −bc, −av, −ao, −ah, −aa, ag, an, au, bb, bi, bp,        bw, cd, ck, cg, bz, bs, bl, be, ax, aq, aj, ac, −ae, −al, −as,        −az, −bg, −bn, −bu, −cb, −ci}    -   {ae, an, aw, bf, bo, bx, cg, ci, bz, bq, bh, ay, ap, ag, −ac,        −al, −au, −bd, −bm, −by, −ce, −ck, −cb, −bs, −bj, −ba, −ar, −ai,        aa, aj, as, bb, bk, bt, cc, cl, cd, bu, bl, bc, at, ak, ab, −ah,        −aq, −az, −bi, −br, −ca, −cj, −cf, −bw, −bn, −be, −av, −am, −ad,        af, ao, ax, bg, bp, by, ch}    -   {af, aq, bb, bm, bx, ci, ce, bt, bi, ax, am, ab, −aj, −au, −bf,        −bq, −cb, −cl, −ca, −bp, −be, −at, −ai, ac, an, ay, bj, bu, cf,        ch, bw, bl, ba, ap, ae, −ag, −ar, −bc, −bn, −by, −cj, −cd, −bs,        −bh, −aw, −al, −aa, ak, av, bg, br, cc, ck, bz, bo, bd, as, ah,        −ad, −ao, −az, −bk, −by, −cg}    -   {ag, at, bg, bt, cg, ce, br, be, ar, ae, −ai, −av, −bi, −by,        −ci, −cc, −bp, −bc, −ap, −ac, ak, ax, bk, bx, ck, ca, bn, ba,        an, aa, −am, −az, −bm, −bz, −cl, −by, −bl, −ay, −al, ab, ao, bb,        bo, cb, cj, bw, bj, aw, aj, −ad, −aq, −bd, −bq, −cd, −ch, −bu,        −bh, −au, −ah, af, as, bf, bs, cf}    -   {ah, aw, bl, ca, ci, bt, be, ap, aa, −ao, −bd, −bs, −ch, −cb,        −bm, −ax, −ai, ag, av, bk, bz, cj, bu, bf, aq, ab, −an, −bc,        −br, −cg, −cc, −bn, −ay, −aj, af, au, bj, by, ck, by, bg, ar,        ac, −am, −bb, −bq, −cf, −cd, −bo, −az, −ak, ae, at, bi, bx, cl,        bw, bh, as, ad, −al, −ba, −bp, −ce}    -   {ai, az, bq, ch, bz, bi, ar, aa, −aq, −bh, −by, −ci, −br, −ba,        −aj, ah, ay, bp, cg, ca, bj, as, ab, −ap, −bg, −bx, −cj, −bs,        −bb, −ak, ag, ax, bo, cf, cb, bk, at, ac, −ao, −bf, −bw, −ck,        −bt, −bc, −al, af, aw, bn, ce, cc, bl, au, ad, −an, −be, −by,        −cl, −bu, −bd, −am, ae, av, bm, cd}    -   {aj, bc, by, cj, bq, ax, ae, −ao, −bh, −ca, −ce, −bl, −as, aa,        at, bm, cf, bz, bg, an, −af, −ay, −br, −ck, −bu, −bb, −ai, ak,        bd, bw, ci, bp, aw, ad, −ap, −bi, −cb, −cd, −bk, −ar, ab, au,        bn, cg, by, bf, am, −ag, −az, −bs, −cl, −bt, −ba, −ah, al, be,        bx, ch, bo, av, ac, −aq, −bj, −cc}    -   {ak, bf, ca, cc, bh, am, −ai, −bd, −by, −ce, −bj, −ao, ag, bb,        bw, cg, bl, aq, −ae, −az, −bu, −ci, −bn, −as, ac, ax, bs, ck,        bp, au, −aa, −av, −bq, −cl, −br, −aw, −ab, at, bo, cj, bt, ay,        ad, −ar, −bm, −ch, −by, −ba, −af, ap, bk, cf, bx, bc, ah, −an,        −bi, −cd, −bz, −be, −aj, al, bg, cb}    -   {al, bi, cf, by, ay, ab, −av, −bs, −ci, −bl, −ao, ai, bf, cc,        by, bb, ae, −as, −bp, −cl, −bo, −ar, af, bc, bz, cb, be, ah,        −ap, −bm, −cj, −br, −au, ac, az, bw, ce, bh, ak, −am, −bj, −cg,        −bu, −ax, −aa, aw, bt, ch, bk, an, −aj, −bg, −cd, −bx, −ba, −ad,        at, bq, ck, bn, aq, −ag, −bd, −ca}    -   {am, bl, ck, bo, ap, −aj, −bi, −ch, −br, −as, ag, bf, ce, bu,        av, −ad, −bc, −cb, −bx, −ay, aa, az, by, ca, bb, ac, −aw, −by,        −cd, −be, −af, at, bs, cg, bh, ai, −aq, −bp, −cj, −bk, −al, an,        bm, cl, bn, ao, −ak, −bj, −ci, −bq, −ar, ah, bg, cf, bt, au,        −ae, −bd, −cc, −bw, −ax, ab, ba, bz}    -   {an, bo, ci, bh, ag, −au, −by, −cb, −ba, aa, bb, cc, bu, at,        −ah, −bi, −cj, −bn, −am, ao, bp, ch, bg, af, −av, −bw, −ca, −az,        ab, bc, cd, bt, as, −ai, −bj, −ck, −bm, −al, ap, bq, cg, bf, ae,        −aw, −bx, −bz, −ay, ac, bd, ce, bs, ar, −aj, −bk, −cl, −bl, −ak,        aq, br, cf, be, ad, −ax, −by}    -   {ao, br, cd, ba, −ac, −bf, −ci, −bm, −aj, at, bw, by, av, −ah,        −bk, −ck, −bh, −ae, ay, cb, bt, aq, −am, −bp, −cf, −bc, aa, bd,        cg, bo, al, −ar, −bu, −ca, −ax, af, bi, cl, bj, ag, −aw, −bz,        −by, −as, ak, bn, ch, be, ab, −bb, −ce, −bq, −an, ap, bs, cc,        az, −ad, −bg, −cj, −bl, −ai, au, bx}    -   {ap, bu, by, at, −al, −bq, −cc, −ax, ah, bm, cg, bb, −ad, −bi,        −ck, −bf, −aa, be, cj, bj, ae, −ba, −cf, −bn, −ai, aw, cb, br,        am, −as, −bx, −by, −aq, ao, bt, bz, au, −ak, −bp, −cd, −ay, ag,        bl, ch, bc, −ac, −bh, −cl, −bg, −ab, bd, ci, bk, af, −az, −ce,        −bo, −aj, av, ca, bs, an, −ar, −bw}    -   {aq, bx, bt, am, −au, −cb, −bp, −ai, ay, cf, bl, ae, −bc, −cj,        −bh, −aa, bg, ck, bd, −ad, −bk, −cg, −az, ah, bo, cc, av, −al,        −bs, −by, −ar, ap, bw, bu, an, −at, −ca, −bq, −aj, ax, ce, bm,        af, −bb, −ci, −bi, −ab, bf, cl, be, −ac, −bj, −ch, −ba, ag, bn,        cd, aw, −ak, −br, −bz, −as, ao, by}    -   {ar, ca, bo, af, −bd, −cl, −bc, ag, bp, bz, aq, −as, −cb, −bn,        −ae, be, ck, bb, −ah, −bq, −by, −ap, at, cc, bm, ad, −bf, −cj,        −ba, ai, br, bx, ao, −au, −cd, −bl, −ac, bg, ci, az, −aj, −bs,        −bw, −an, av, ce, bk, ab, −bh, −ch, −ay, ak, bt, by, am, −aw,        −cf, −bj, −aa, bi, cg, ax, −al, −bu}    -   {as, cd, bj, −ab, −bm, −ca, −ap, av, cg, bg, −ae, −bp, −bx, −am,        ay, cj, bd, −ah, −bs, −bu, −aj, bb, cl, ba, −ak, −by, −br, −ag,        be, ci, ax, −an, −by, −bo, −ad, bh, cf, au, −aq, −cb, −bl, −aa,        bk, cc, ar, −at, −ce, −bi, ac, bn, bz, ao, −aw, −ch, −bf, af,        bq, bw, al, −az, −ck, −bc, ai, bt}    -   {at, cg, be, −ai, −by, −bp, −ac, bk, ca, an, −az, −cl, −ay, ao,        cb, bj, −ad, −bq, −bu, −ah, bf, cf, as, −au, −ch, −bd, aj, bw,        bo, ab, −bl, −bz, −am, ba, ck, ax, −ap, −cc, −bi, ae, br, bt,        ag, −bg, −ce, −ar, av, ci, bc, −ak, −bx, −bn, −aa, bm, by, al,        −bb, −cj, −aw, aq, cd, bh, −af, −bs}    -   {au, cj, az, −ap, −ce, −be, ak, bz, bj, −af, −bu, −bo, aa, bp,        bt, ae, −bk, −by, −aj, bf, cd, ao, −ba, −ci, −at, av, ck, ay,        −aq, −cf, −bd, al, ca, bi, −ag, −by, −bn, ab, bq, bs, ad, −bl,        −bx, −ai, bg, cc, an, −bb, −ch, −as, aw, cl, ax, −ar, −cg, −bc,        am, cb, bh, −ah, −bw, −bm, ac, br}    -   {av, cl, au, −aw, −ck, −at, ax, cj, as, −ay, −ci, −ar, az, ch,        aq, −ba, −cg, −ap, bb, cf, ao, −bc, −ce, −an, bd, cd, am, −be,        −cc, −al, bf, cb, ak, −bg, −ca, −aj, bh, bz, ai, −bi, −by, −ah,        bj, bx, ag, −bk, −bw, −af, bl, by, ae, −bm, −bu, −ad, bn, bt,        ac, −bo, −bs, −ab, bp, br, aa, −bq}    -   {aw, ci, ap, −bd, −cb, −ai, bk, bu, ab, −br, −bn, af, by, bg,        −am, −cf, −az, at, cl, as, −ba, −ce, −al, bh, bx, ae, −bo, −bq,        ac, by, bj, −aj, −cc, −bc, aq, cj, av, −ax, −ch, −ao, be, ca,        ah, −bl, −bt, −aa, bs, bm, −ag, −bz, −bf, an, cg, ay, −au, −ck,        −ar, bb, cd, ak, −bi, −bw, −ad, bp}    -   {ax, cf, ak, −bk, −bs, ac, bx, bf, −ap, −ck, −as, bc, ca, af,        −bp, −bn, ah, cc, ba, −au, −ci, −an, bh, by, aa, −bu, −bi, am,        ch, av, −az, −cd, −ai, bm, bq, −ae, −bz, −bd, ar, cl, aq, −be,        −by, −ad, br, bl, −aj, −ce, −ay, aw, cg, al, −bj, −bt, ab, bw,        bg, −ao, −cj, −at, bb, cb, ag, −bo}    -   {ay, cc, af, −br, −bj, an, ck, aq, −bg, −bu, ac, bz, bb, −av,        −cf, −ai, bo, bm, −ak, −ch, −at, bd, bx, aa, −bw, −be, as, ci,        al, −bl, −bp, ah, ce, aw, −ba, −ca, −ad, bt, bh, −ap, −cl, −ao,        bi, bs, −ae, −cb, −az, ax, cd, ag, −bq, −bk, am, cj, ar, −bf,        −by, ab, by, bc, −au, −cg, −aj, bn}    -   {az, bz, aa, −by, −ba, ay, ca, ab, −bx, −bb, ax, cb, ac, −bw,        −bc, aw, cc, ad, −by, −bd, av, cd, ae, −bu, −be, au, ce, af,        −bt, −bf, at, cf, ag, −bs, −bg, as, cg, ah, −br, −bh, ar, ch,        ai, −bq, −bi, aq, ci, aj, −bp, −bj, ap, cj, ak, −bo, −bk, ao,        ck, al, −bn, −bl, an, cl, am, −bm}    -   {ba, bw, −ae, −cf, −ar, bj, bn, −an, −cj, −ai, bs, be, −aw, −ca,        aa, cb, av, −bf, −br, aj, ck, am, −bo, −bi, as, ce, ad, −bx,        −az, bb, by, −af, −cg, −aq, bk, bm, −ao, −ci, −ah, bt, bd, −ax,        −bz, ab, cc, au, −bg, −bq, ak, cl, al, −bp, −bh, at, cd, ac,        −by, −ay, bc, bu, −ag, −ch, −ap, bl}    -   {bb, bt, −aj, −cl, −ai, bu, ba, −bc, −bs, ak, ck, ah, −by, −az,        bd, br, −al, −cj, −ag, bw, ay, −be, −bq, am, ci, af, −bx, −ax,        bf, bp, −an, −ch, −ae, by, aw, −bg, −bo, ao, cg, ad, −bz, −av,        bh, bn, −ap, −cf, −ac, ca, au, −bi, −bm, aq, ce, ab, −cb, −at,        bj, bl, −ar, −cd, −aa, cc, as, −bk}    -   {bc, bq, −ao, −ce, aa, cf, an, −br, −bb, bd, bp, −ap, −cd, ab,        cg, am, −bs, −ba, be, bo, −aq, −cc, ac, ch, al, −bt, −az, bf,        bn, −ar, −cb, ad, ci, ak, −bu, −ay, bg, bm, −as, −ca, ae, cj,        aj, −by, −ax, bh, bl, −at, −bz, af, ck, ai, −bw, −aw, bi, bk,        −au, −by, ag, cl, ah, −bx, −av, bj}    -   {bd, bn, −at, −bx, aj, ch, aa, −cg, −ak, bw, au, −bm, −be, bc,        bo, −as, −by, ai, ci, ab, −cf, −al, by, av, −bl, −bf, bb, bp,        −ar, −bz, ah, cj, ac, −ce, −am, bu, aw, −bk, −bg, ba, bq, −aq,        −ca, ag, ck, ad, −cd, −an, bt, ax, −bj, −bh, az, br, −ap, −cb,        af, cl, ae, −cc, −ao, bs, ay, −bi}    -   {be, bk, −ay, −bq, as, bw, −am, −cc, ag, ci, −aa, −cj, −af, cd,        al, −bx, −ar, br, ax, −bl, −bd, bf, bj, −az, −bp, at, by, −an,        −cb, ah, ch, −ab, −ck, −ae, ce, ak, −by, −aq, bs, aw, −bm, −bc,        bg, bi, −ba, −bo, au, bu, −ao, −ca, ai, cg, −ac, −cl, −ad, cf,        aj, −bz, −ap, bt, av, −bn, −bb, bh}    -   {bf, bh, −bd, −bj, bb, bl, −az, −bn, ax, bp, −av, −br, at, bt,        −ar, −by, ap, bx, −an, −bz, al, cb, −aj, −cd, ah, cf, −af, −ch,        ad, cj, −ab, −cl, −aa, ck, ac, −ci, −ae, cg, ag, −ce, −ai, cc,        ak, −ca, −am, by, ao, −bw, −aq, bu, as, −bs, −au, bq, aw, −bo,        −ay, bm, ba, −bk, −bc, bi, be, −bg}    -   {bg, be, −bi, −bc, bk, ba, −bm, −ay, bo, aw, −bq, −au, bs, as,        −bu, −aq, bw, ao, −by, −am, ca, ak, −cc, −ai, ce, ag, −cg, −ae,        ci, ac, −ck, −aa, cl, −ab, −cj, ad, ch, −af, −cf, ah, cd, −aj,        −cb, al, bz, −an, −bx, ap, by, −ar, −bt, at, br, −av, −bp, ax,        bn, −az, −bl, bb, bj, −bd, −bh, bf}    -   {bh, bb, −bn, −av, bt, ap, −bz, −aj, cf, ad, −cl, ac, cg, −ai,        −ca, ao, bu, −au, −bo, ba, bi, −bg, −bc, bm, aw, −bs, −aq, by,        ak, −ce, −ae, ck, −ab, −ch, ah, cb, −an, −by, at, bp, −az, −bj,        bf, bd, −bl, −ax, br, ar, −bx, −al, cd, af, −cj, aa, ci, −ag,        −cc, am, bw, −as, −bq, ay, bk, −be}    -   {bi, ay, −bs, −ao, cc, ae, −cl, af, cb, −ap, −br, az, bh, −bj,        −ax, bt, an, −cd, −ad, ck, −ag, −ca, aq, bq, −ba, −bg, bk, aw,        −bu, −am, ce, ac, −cj, ah, bz, −ar, −bp, bb, bf, −bl, −av, by,        al, −cf, −ab, ci, −ai, −by, as, bo, −bc, −be, bm, au, −bw, −ak,        cg, aa, −ch, aj, bx, −at, −bn, bd}    -   {bj, av, −bx, −ah, cl, −ag, −by, au, bk, −bi, −aw, bw, ai, −ck,        af, bz, −at, −bl, bh, ax, −by, −aj, cj, −ae, −ca, as, bm, −bg,        −ay, bu, ak, −ci, ad, cb, −ar, −bn, bf, az, −bt, −al, ch, −ac,        −cc, aq, bo, −be, −ba, bs, am, −cg, ab, cd, −ap, −bp, bd, bb,        −br, −an, cf, −aa, −ce, ao, bq, −bc}    -   {bk, as, −cc, −aa, cd, −ar, −bl, bj, at, −cb, −ab, ce, −aq, −bm,        bi, au, −ca, −ac, cf, −ap, −bn, bh, av, −bz, −ad, cg, −ao, −bo,        bg, aw, −by, −ae, ch, −an, −bp, bf, ax, −bx, −af, ci, −am, −bq,        be, ay, −bw, −ag, cj, −al, −br, bd, az, −by, −ah, ck, −ak, −bs,        bc, ba, −bu, −ai, cl, −aj, −bt, bb}    -   {bl, ap, −ch, ag, bu, −bc, −ay, by, ac, −cd, at, bh, −bp, −al,        cl, −ak, −bq, bg, au, −cc, ab, bz, −ax, −bd, bt, ah, −ci, ao,        bm, −bk, −aq, cg, −af, −by, bb, az, −bx, −ad, ce, −as, −bi, bo,        am, −ck, aj, br, −bf, −av, cb, −aa, −ca, aw, be, −bs, −ai, cj,        −an, −bn, bj, ar, −cf, ae, bw, −ba}    -   {bm, am, −cl, an, bl, −bn, −al, ck, −ao, −bk, bo, ak, −cj, ap,        bj, −bp, −aj, ci, −aq, −bi, bq, ai, −ch, ar, bh, −br, −ah, cg,        −as, −bg, bs, ag, −cf, at, bf, −bt, −af, ce, −au, −be, bu, ae,        −cd, av, bd, −by, −ad, cc, −aw, −bc, bw, ac, −cb, ax, bb, −bx,        −ab, ca, −ay, −ba, by, aa, −bz, az}    -   {bn, aj, −cg, au, bc, −by, ab, by, −bf, −ar, cj, −am, −bk, bq,        ag, −cd, ax, az, −cb, ae, bs, −bi, −ao, cl, −ap, −bh, bt, ad,        −ca, ba, aw, −ce, ah, bp, −bl, −al, ci, −as, −be, bw, aa, −bx,        bd, at, −ch, ak, bm, −bo, −ai, cf, −av, −bb, bz, −ac, −bu, bg,        aq, −ck, an, bj, −br, −af, cc, −ay}    -   {bo, ag, −cb, bb, at, −cj, ao, bg, −bw, ab, bt, −bj, −al, cg,        −aw, −ay, ce, −aj, −bl, br, ad, −by, be, aq, −cl, ar, bd, −bz,        ae, bq, −bm, −ai, cd, −az, −av, ch, −am, −bi, bu, aa, −by, bh,        an, −ci, au, ba, −cc, ah, bn, −bp, −af, ca, −bc, −as, ck, −ap,        −bf, bx, −ac, −bs, bk, ak, −cf, ax}    -   {bp, ad, −bw, bi, ak, −cd, bb, ar, −ck, au, ay, −cg, an, bf,        −bz, ag, bm, −bs, −aa, bt, −bl, −ah, ca, −be, −ao, ch, −ax, −av,        cj, −aq, −bc, cc, −aj, −bj, by, −ac, −bq, bo, ae, −bx, bh, al,        −ce, ba, as, −cl, at, az, −cf, am, bg, −by, af, bn, −br, −ab,        bu, −bk, −ai, cb, −bd, −ap, ci, −aw}    -   {bq, aa, −br, bp, ab, −bs, bo, ac, −bt, bn, ad, −bu, bm, ae,        −by, bl, af, −bw, bk, ag, −bx, bj, ah, −by, bi, ai, −bz, bh, aj,        −ca, bg, ak, −cb, bf, al, −cc, be, am, −cd, bd, an, −ce, bc, ao,        −cf, bb, ap, −cg, ba, aq, −ch, az, ar, −ci, ay, as, −cj, ax, at,        −ck, aw, au, −cl, av}    -   {br, −ac, −bm, bw, −ah, −bh, cb, −am, −bc, cg, −ar, −ax, cl,        −aw, −as, ch, −bb, −an, cc, −bg, −ai, bx, −bl, −ad, bs, −bq, ab,        bn, −by, ag, bi, −ca, al, bd, −cf, aq, ay, −ck, av, at, −ci, ba,        ao, −cd, bf, aj, −by, bk, ae, −bt, bp, −aa, −bo, bu, −af, −bj,        bz, −ak, −be, ce, −ap, −az, cj, −au}    -   {bs, −af, −bh, cd, −aq, −aw, cj, −bb, −al, by, −bm, −aa, bn,        −bx, ak, bc, −ci, av, ar, −ce, bg, ag, −bt, br, −ae, −bi, cc,        −ap, −ax, ck, −ba, −am, bz, −bl, −ab, bo, −bw, aj, bd, −ch, au,        as, −cf, bf, ah, −bu, bq, −ad, −bj, cb, −ao, −ay, cl, −az, −an,        ca, −bk, −ac, bp, −by, ai, be, −cg, at}    -   {bt, −ai, −bc, ck, −az, −al, bw, −bq, af, bf, −ch, aw, ao, −bz,        bn, −ac, −bi, ce, −at, −ar, cc, −bk, −aa, bl, −cb, aq, au, −cf,        bh, ad, −bo, by, −an, −ax, ci, −be, −ag, br, −by, ak, ba, −cl,        bb, aj, −bu, bs, −ah, −bd, cj, −ay, −am, bx, −bp, ae, bg, −cg,        av, ap, −ca, bm, −ab, −bj, cd, −as}    -   {bu, −al, −ax, cg, −bi, −aa, bj, −cf, aw, am, −by, bt, −ak, −ay,        ch, −bh, −ab, bk, −ce, av, an, −bw, bs, −aj, −az, ci, −bg, −ac,        bl, −cd, au, ao, −bx, br, −ai, −ba, cj, −bf, −ad, bm, −cc, at,        ap, −by, bq, −ah, −bb, ck, −be, −ae, bn, −cb, as, aq, −bz, bp,        −ag, −bc, cl, −bd, −af, bo, −ca, ar}    -   {by, −ao, −as, bz, −br, ak, aw, −cd, bn, −ag, −ba, ch, −bj, ac,        be, −cl, bf, ab, −bi, ci, −bb, of bm, −ce, ax, aj, −bq, ca, −at,        −an, bu, −bw, ap, ar, −by, bs, −al, −av, cc, −bo, ah, az, −cg,        bk, −ad, −bd, ck, −bg, −aa, bh, −cj, bc, ae, −bl, cf, −ay, −ai,        bp, −cb, au, am, −bt, bx, −aq}    -   {bw, −ar, −an, bs, −ca, av, aj, −bo, ce, −az, −af, bk, −ci, bd,        ab, −bg, cl, −bh, ac, bc, −ch, bl, −ag, −ay, cd, −bp, ak, au,        −bz, bt, −ao, −aq, by, −bx, as, am, −br, cb, −aw, −ai, bn, −cf,        ba, ae, −bj, cj, −be, −aa, bf, −ck, bi, −ad, −bb, cg, −bm, ah,        ax, −cc, bq, −al, −at, by, −bu, ap}    -   {bx, −au, −ai, bl, −cj, bg, −ad, −az, cc, −bs, ap, an, −bq, ce,        −bb, −ab, be, −ch, bn, −ak, −as, by, −bz, aw, ag, −bj, cl, −bi,        af, ax, −ca, bu, −ar, −al, bo, −cg, bd, −aa, −bc, cf, −bp, am,        aq, −bt, cb, −ay, −ae, bh, −ck, bk, −ah, −av, by, −bw, at, aj,        −bm, ci, −bf, ac, ba, −cd, br, −ao}    -   {by, −ax, −ad, be, −cf, br, −aq, −ak, bl, −cl, bk, −aj, −ar, bs,        −ce, bd, −ac, −ay, bz, −bx, aw, ae, −bf, cg, −bq, ap, al, −bm,        ck, −bj, ai, as, −bt, cd, −bc, ab, az, −ca, bw, −av, −af, bg,        −ch, bp, −ao, −am, bn, −cj, bi, −ah, −at, bu, −cc, bb, −aa, −ba,        cb, −by, au, ag, −bh, ci, −bo, an}    -   {bz, −ba, ab, ax, −bw, cc, −bd, ae, au, −bt, cf, −bg, ah, ar,        −bq, ci, −bj, ak, ao, −bn, cl, −bm, an, al, −bk, cj, −bp, aq,        ai, −bh, cg, −bs, at, af, −be, cd, −by, aw, ac, −bb, ca, −by,        az, −aa, −ay, bx, −cb, bc, −ad, −av, bu, −ce, bf, −ag, −as, br,        −ch, bi, −aj, −ap, bo, −ck, bl, −am}    -   {ca, −bd, ag, aq, −bn, ck, −bq, at, ad, −ba, bx, −cd, bg, −aj,        −an, bk, −ch, bt, −aw, −aa, ax, −bu, cg, −bj, am, ak, −bh, ce,        −bw, az, −ac, −au, br, −cj, bm, −ap, −ah, be, −cb, bz, −bc, af,        ar, −bo, cl, −bp, as, ae, −bb, by, −cc, bf, −ai, −ao, bl, −ci,        bs, −av, −ab, ay, −by, cf, −bi, al}    -   {cb, −bg, al, aj, −be, bz, −cd, bi, −an, −ah, bc, −bx, cf, −bk,        ap, af, −ba, by, −ch, bm, −ar, −ad, ay, −bt, cj, −bo, at, ab,        −aw, br, −cl, bq, −av, aa, au, −bp, ck, −bs, ax, −ac, −as, bn,        −ci, bu, −az, ae, aq, −bl, cg, −bw, bb, −ag, −ao, bj, −ce, by,        −bd, ai, am, −bh, cc, −ca, bf, −ak}    -   {cc, −bj, aq, ac, −av, bo, −ch, bx, −be, al, ah, −ba, bt, −cl,        bs, −az, ag, am, −bf, by, −cg, bn, −au, ab, ar, −bk, cd, −cb,        bi, −ap, −ad, aw, −bp, ci, −bw, bd, −ak, −ai, bb, −bu, ck, −br,        ay, −af, −an, bg, −bz, cf, −bm, at, −aa, −as, bl, −ce, ca, −bh,        ao, ae, −ax, bq, −cj, by, −bc, aj}    -   {cd, −bm, av, −ae, −am, bd, −bu, cl, −by, be, −an, −ad, au, −bl,        cc, −ce, bn, −aw, af, al, −bc, bt, −ck, bw, −bf, ao, ac, −at,        bk, −cb, cf, −bo, ax, −ag, −ak, bb, −bs, cj, −bx, bg, −ap, −ab,        as, −bj, ca, −cg, bp, −ay, ah, aj, −ba, br, −ci, by, −bh, aq,        aa, −ar, bi, −bz, ch, −bq, az, −ai}    -   {ce, −bp, ba, −al, −ad, as, −bh, bw, −cl, bx, −bi, at, −ae, −ak,        az, −bo, cd, −cf, bq, −bb, am, ac, −ar, bg, −by, ck, −by, bj,        −au, af, aj, −ay, bn, −cc, cg, −br, bc, −an, −ab, aq, −bf, bu,        −cj, bz, −bk, av, −ag, −ai, ax, −bm, cb, −ch, bs, −bd, ao, aa,        −ap, be, −bt, ci, −ca, bl, −aw, ah}    -   {cf, −bs, bf, −as, af, ah, −au, bh, −bu, ch, −cd, bq, −bd, aq,        −ad, −aj, aw, −bj, bw, −cj, cb, −bo, bb, −ao, ab, al, −ay, bl,        −by, cl, −bz, bm, −az, am, aa, −an, ba, −bn, ca, −ck, bx, −bk,        ax, −ak, −ac, ap, −bc, bp, −cc, ci, −by, bi, −av, ai, ae, −ar,        be, −br, ce, −cg, bt, −bg, at, −ag}    -   {cg, −by, bk, −az, ao, −ad, −ah, as, −bd, bo, −bz, ck, −cc, br,        −bg, av, −ak, −aa, al, −aw, bh, −bs, cd, −cj, by, −bn, bc, −ar,        ag, ae, −ap, ba, −bl, bw, −ch, cf, −bu, bj, −ay, an, −ac, −ai,        at, −be, bp, −ca, cl, −cb, bq, −bf, au, −aj, −ab, am, −ax, bi,        −bt, ce, −ci, bx, −bm, bb, −aq, af}    -   {ch, −by, bp, −bg, ax, −ao, af, ad, −am, av, −be, bn, −bw, cf,        −cj, ca, −br, bi, −az, aq, −ah, −ab, ak, −at, bc, −bl, bu, −cd,        cl, −cc, bt, −bk, bb, −as, aj, −aa, −ai, ar, −ba, bj, −bs, cb,        −ck, ce, −by, bm, −bd, au, −al, ac, ag, −ap, ay, −bh, bq, −bz,        ci, −cg, bx, −bo, bf, −aw, an, −ae}    -   {ci, −cb, bu, −bn, bg, −az, as, −al, ae, ac, −aj, aq, −ax, be,        −bl, bs, −bz, cg, −ck, cd, −bw, bp, −bi, bb, −au, an, −ag, −aa,        ah, −ao, av, −bc, bj, −bq, bx, −ce, cl, −cf, by, −br, bk, −bd,        aw, −ap, ai, −ab, −af, am, −at, ba, −bh, bo, −by, cc, −cj, ch,        −ca, bt, −bm, bf, −ay, ar, −ak, ad}    -   {cj, −ce, bz, −bu, bp, −bk, bf, −ba, av, −aq, al, −ag, ab, ad,        −ai, an, −as, ax, −bc, bh, −bm, br, −bw, cb, −cg, cl, −ch, cc,        −bx, bs, −bn, bi, −bd, ay, −at, ao, −aj, ae, aa, −af, ak, −ap,        au, −az, be, −bj, bo, −bt, by, −cd, ci, −ck, cf, −ca, by, −bq,        bl, −bg, bb, −aw, ar, −am, ah, −ac}    -   {ck, −ch, ce, −cb, by, −by, bs, −bp, bm, −bj, bg, −bd, ba, −ax,        au, −ar, ao, −al, ai, −af, ac, aa, −ad, ag, −aj, am, −ap, as,        −av, ay, −bb, be, −bh, bk, −bn, bq, −bt, bw, −bz, cc, −cf, ci,        −cl, cj, −cg, cd, −ca, bx, −bu, br, −bo, bl, −bi, bf, −bc, az,        −aw, at, −aq, an, −ak, ah, −ae, ab}    -   {cl, −ck, cj, −ci, ch, −cg, cf, −ce, cd, −cc, cb, −ca, bz, −by,        bx, −bw, by, −bu, bt, −bs, br, −bq, bp, −bo, bn, −bm, bl, −bk,        bj, −bi, bh, −bg, bf, −be, bd, −bc, bb, −ba, az, −ay, ax, −aw,        av, −au, at, −as, ar, −aq, ap, −ao, an, −am, al, −ak, aj, −ai,        ah, −ag, af, −ae, ad, −ac, ab, −aa}

The variables a, b c, . . . aa, ab, ac, . . . , cl can be derived basedon the formulations of DST-7 shown in Table, and their values can bedifferent for different sizes of DST-7. For example, the value of “a”can be different for 4-point DST-7 and 8-point DST-7. To avoid floatingpoint operations, similar to the DCT-2 cores used in HEVC, the transformcore of DST-7 may be scaled by a pre-defined factor, e.g., 64 √{squareroot over (N)} (corresponds to 8-bit transform core) or 256·√{squareroot over (N)} (corresponds to 10-bit transform core), and rounded tothe nearest integer, or further tuned by an offset, e.g., +1/−1.

In an exemplary design of AMT, when AMT is applied, all primarytransform cores, including DCT-2, may be represented by 10-bit integers,i.e., a 10-bit transform core. However, it may be the case that at leastone of the primary transform cores (but not all) is represented by10-bit integers.

The primary transform in VVC may apply both 8-bit (e.g., used for allother transform sizes except for 64-point and larger) and 10-bit (e.g.,used for 64-point transform and larger) transform cores. In thisapplication, the 2-, 4-, 8-, 16- and 32-point DCT-2 transform corecannot be extracted from, for example, a 64-point DCT-2 transform core,which inevitably increases the cost of applying primary transformbecause a separate 64-point transform core would need extra storage andlogic for implementation. Comparing to 10-bit primary transform cores,the 8-bit primary transform core reduces the complexity because of thereduced memory for storing transform cores and reduced input bit-widthof multiplier/add/sub operations.

Thus, the disclosure provides a set of 8-bit primary transform cores forall transform size types applied in an exemplary AMT including DCT-2,DCT-5, DCT-8, DST-1 and DST-7. The following exemplary proposed methodsmay be used separately or combined in any order.

Referring to FIG. 5 , in a first, simple example, a method of utilizingan 8-bit primary transform core matrix to decode compressed video orimage data or encode uncompressed video or image data may include: Step501: receiving information regarding a target data block for encoding ordecoding. The target data block for encoding or decoding being one of: acompressed video or image data block or an uncompressed video or imagedata block. The method of FIG. 5 may also include a Step 502 thatincludes determining, for the encoding or decoding of the target datablock, whether to use a first transform core matrix that is a DST-7primary transform core matrix which is larger than or equal to 8-pointand represented by 8-bits. The method according to the aspect of thedisclosure may further include a Step 503 that includes, when a resultof the determination is to use the first transform core matrix: (i)deriving the 8-bit DST-7 primary transform core matrix based on a set ofM unique numbers, and (ii) causing or transmitting information thatcauses the target data block for encoding or decoding to be encoded ordecoded using the derived 8-bit DST-7 primary transform core matrix. Themethod of FIG. 5 may also include a Step 504 that includes, when aresult of the determination is to not use the first transform corematrix: (i) deriving a second primary transform core matrix based on thesame set of M unique numbers, and (ii) causing or transmittinginformation that causes the target data block for encoding or decodingto be encoded or decoded using the derived second primary transform corematrix, wherein the second primary transform core matrix is one ofDCT-8, DST-1 and DCT-5. The method illustrated in FIG. 5 may alsoinclude that the second primary transform core matrix is one of DCT-8,DST-1 and DCT-5.

The method illustrated in FIG. 5 may also include processing Steps501-504 being performed by one or more processors. The methodillustrated in FIG. 5 may also include storing, by the one or moreprocessors in a memory, the 8-bit primary transform core matrix inadvance before or during the performing of the Steps 501-504. The methodillustrated in FIG. 5 may also include that the second primary transformcore matrix is an 8-bit DCT-8 primary transform core matrix. The methodillustrated in FIG. 5 may also include that when deriving the N-point8-bit DCT-8 primary transform core matrix or the N-point 8-bit DST-7transform core matrix, the set of M unique numbers which are used toconstruct the N-point DCT-8 core matrix is the same set of M uniquenumbers which construct the N-point DST-7 core matrix. In oneembodiment, M equals to N.

The method according to FIG. 5 may further comprise selecting atransform subset from one or more transform subsets, the transformsubsets each including at least two transforms, the selected transformsubset including DST-7 and DCT-8, and for each of a horizontal transformand a vertical transform, one of the at least two transform candidatesin the selected transform subset is selected based on being explicitlysignaled with flags. The method illustrated in FIG. 5 may also includethat the second primary transform core matrix is an 8-bit DCT-1 primarytransform core matrix. The method illustrated in FIG. 5 may also includethat when deriving a 2N-point DST-1 transform core, the set of M uniquenumbers which construct the 2N-point DST-1 core matrix is the same setof M unique numbers which construct the N-point DST-7. In oneembodiment, M equals to N.

The method illustrated in FIG. 5 may also include that for interprediction residual, only one transform set, which consists of DST-7 andDCT-8, is used for all inter modes and for both horizontal and verticaltransforms.

The method illustrated in FIG. 5 may also include that the 8-bit,N-point primary transform core matrix, which is composed by M uniquenumbers, is derived by the following steps: deriving an initialtransform core matrix T1 by scaling an original transform core which iscomposed by floating-point numbers T0 using a scaling factor 2B−2×log2(N) such that T1=T0×2^(B-2)×log₂(N), where B=8, and the row vectors ofT0, which are basis vectors, are orthogonal to each other, and the normof each row vector is 1; deriving an adjusted transform core matrix T2by adding offsets on a selected set of the M unique numbers which areused to construct the adjusted primary transform core matrix T2;calculating a cost value C as the sum of absolute values of all elementsof (T1*T1^(T)−2^(2B-4)×log₂ (N)²×I), where I is an N×N identity matrix;and updating the adjusted transform matrix T2 by adding the calculatedoffset values to T2, which minimizes the cost value to T2, which isoutput as the 8-bit, N-point primary transform core matrix. Although itshould be noted that zero, only one or more of, or all of these substeps(above in this same paragraph) may be performed.

The method of FIG. 5 may further include that after minimizing the costvalue C by trying all possible combinations of offset values on aselected set of M unique numbers, T2 is output as the 8-bit, N-pointprimary transform core matrix.

The method of FIG. 5 may further include that the first transform corematrix is an 8-point, 8-bit DST-7 primary transform core matrix that isconstructed using eight integers that include: a, b, c, d, e, f, g and hplus their respective sign changes, and the eight integers {a, b, c, d,e, f, g, h} are equal to {17, 32, 46, 60, 71, 78, 85, 86}.

The method of FIG. 5 may further include that the first primarytransform core matrix is a 16-point, 8-bit DST-7 primary transform corematrix that is constructed using eight integers that include: a, b, c,d, e, f, g, h, i, j, k, l, m, n, o and p plus their respective signchanges, and the sixteen integers {a, b, c, d, e, f, g, h, i, j, k, l,m, n, o, p} are equal to {9, 17, 25, 33, 41, 49, 56, 62, 66, 72, 77, 81,83, 87, 89, 90}.

The method of FIG. 5 may further include that the first transform corematrix is a 32-point, 8-bit DST-7 primary transform core matrix that isconstructed using thirty-two (32) integers. The thirty-two integers mayinclude: a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u,v, w, x, y, z, A, B, C, D, E, and F plus their respective sign changes,and the thirty-two integers {a, b, c, d, e, f, g, h, i, j, k, l, m, n,o, p, q, r, s, t, u, v, w, x, y, z, A, B, C, D, E, F} may be equal to{4, 9, 13, 17, 21, 26, 30, 34, 38, 42, 45, 50, 53, 56, 60, 63, 66, 68,72, 74, 77, 78, 80, 82, 84, 85, 86, 88, 88, 89, 90, 90}.

The method of FIG. 5 may further include that the first transform corematrix is a 64-point, 8-bit DST-7 primary transform core matrix that isconstructed using sixty-four (64) integers. The sixty-four integers mayinclude: aa, ab, ac, ad, ae, af, ag, ah, ai, aj, ak, al, am, an, ao, ap,aq, ar, as, at, au, av, aw, ax, ay, az, ba, bb, bc, bd, be, bf, bg, bh,bi, bj, bk, bl, bm, bn, bo, bp, bq, br, bs, bt, bu, by, bw, bx, by, bz,ca, cb, cc, cd, ce, cf, cg, ch, ci, cj, ck, and cl plus their respectivesign changes, and the sixty-four integers {aa, ab, ac, ad, ae, af, ag,ah, ai, aj, ak, al, am, an, ao, ap, aq, ar, as, at, au, av, aw, ax, ay,az, ba, bb, bc, bd, be, bf, bg, bh, bi, bj, bk, bl, bm, bn, bo, bp, bq,br, bs, bt, bu, by, bw, bx, by, bz, ca, cb, cc, cd, ce, cf, cg, ch, ci,cj, ck, cl} may be equal to {2, 5, 7, 8, 11, 13, 15, 17, 20, 22, 23, 26,29, 30, 32, 34, 37, 38, 41, 42, 44, 46, 48, 49, 52, 53, 55, 58, 59, 60,62, 64, 65, 66, 68, 69, 72, 72, 74, 73, 76, 77, 78, 79, 81, 80, 82, 83,85, 84, 85, 86, 87, 87, 88, 89, 89, 90, 89, 90, 90, 90, 90, 90}.

The method of FIG. 5 may further include that the second transform corematrix is an 8-point, 8-bit DCT-8 primary transform core matrix that isconstructed using eight (8) integers. The eight integers may include: a,b, c, d, e, f, g and h plus their respective sign changes, and the eightintegers {a, b, c, d, e, f, g, h} may be equal to {86, 85, 78, 71, 60,46, 32, 17}.

The method of FIG. 5 may further include that the second transform corematrix is a 16-point, 8-bit DCT-8 primary transform core matrix that isconstructed using sixteen (16) integers. The sixteen integers mayinclude: a, b, c, d, e, f, g, h, i, j, k, l, m, n, o and p plus theirrespective sign changes, and the sixteen integers {a, b, c, d, e, f, g,h, i, j, k, l, m, n, o, p} may be equal to {90, 89, 87, 83, 81, 77, 72,66, 62, 56, 49, 41, 33, 25, 17, 9}.

The method of FIG. 5 may further include that the second transform corematrix is a 32-point, 8-bit DCT-8 primary transform core matrix. Thesecond transform core matrix may be constructed using thirty-two (32)integers. The thirty-two integers may include: a, b, c, d, e, f, g, h,i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z, A, B, C, D, E, andF plus their respective sign changes, and the thirty-two integers {a, b,c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z,A, B, C, D, E, F} may be equal to {90, 90, 89, 88, 88, 86, 85, 84, 82,80, 78, 77, 74, 72, 68, 66, 63, 60, 56, 53, 50, 45, 42, 38, 34, 30, 26,21, 17, 13, 9, 4}.

The method of FIG. 5 may further include that the second transform corematrix is a 64-point, 8-bit DCT-8 primary transform core matrix that isconstructed using sixty-four (64) integers. The sixty-four integers mayinclude: aa, ab, ac, ad, ae, af, ag, ah, ai, aj, ak, al, am, an, ao, ap,aq, ar, as, at, au, av, aw, ax, ay, az, ba, bb, bc, bd, be, bf, bg, bh,bi, bj, bk, bl, bm, bn, bo, bp, bq, br, bs, bt, bu, by, bw, bx, by, bz,ca, cb, cc, cd, ce, cf, cg, ch, ci, cj, ck, and cl plus their respectivesign changes, and the sixty-four integers {aa, ab, ac, ad, ae, af, ag,ah, ai, aj, ak, al, am, an, ao, ap, aq, ar, as, at, au, av, aw, ax, ay,az, ba, bb, bc, bd, be, bf, bg, bh, bi, bj, bk, bl, bm, bn, bo, bp, bq,br, bs, bt, bu, by, bw, bx, by, bz, ca, cb, cc, cd, ce, cf, cg, ch, ci,cj, ck, cl} may be equal to {90, 90, 90, 90, 90, 89, 90, 89, 89, 88, 87,87, 86, 85, 84, 85, 83, 82, 80, 81, 79, 78, 77, 76, 73, 74, 72, 72, 69,68, 66, 65, 64, 62, 60, 59, 58, 55, 53, 52, 49, 48, 46, 44, 42, 41, 38,37, 34, 32, 30, 29, 26, 23, 22, 20, 17, 15, 13, 11, 8, 7, 5, 2}.

The method illustrated in FIG. 5 may also include that the 8-bit,N-point primary transform core matrix, which is composed by M uniquenumbers, is derived by the following steps illustrated in FIG. 6 . FIG.6 illustrates a Step 601 that includes deriving an initial transformcore matrix by scaling an original transform core using a scalingfactor. For example, the Step 601 may include deriving an initialtransform core matrix T1 by scaling an original transform core which iscomposed by floating-point numbers T0 using a scaling factor 2B−2×log2(N) such that T1=T0×2^(B-2)×log₂(N), where B=8, and the row vectors ofT0, which are basis vectors, are orthogonal to each other, and the normof each row vector is 1.

The method of FIG. 6 may also include a Step 602 that includes derivingan adjusted transform core matrix. For example, Step 602 may includederiving an adjusted transform core matrix (T2) by adding offsets on aselected set of the M unique numbers which are used to construct theadjusted primary transform core matrix T2.

The method of FIG. 6 may also include a Step 603 that includescalculating a cost value C as the sum of absolute values of allelements. For example, the Step 603 may include calculating a cost valueC as the sum of absolute values of all elements of(T1*T1^(T)−2^(2B-4)×log₂(N)²×I), where I is an N×N identity matrix.

The method of FIG. 6 may also include a Step 604 that includes updatingthe adjusted transform matrix. For example, the Step 604 may includeupdating the adjusted transform matrix T2 by adding the calculatedoffset values to T2, which minimizes the cost value to T2, which may beoutput as the 8-bit, N-point primary transform core matrix.

In the method of FIG. 6 , the values of the offsets for each of the Munique numbers may include one or more of the following values: −2, −1,0, +1, +2. Moreover, the method of FIG. 6 may also include afterminimizing the cost value C by trying all possible combinations ofoffset values on a selected set of M unique numbers, outputting T2 asthe 8-bit, N-point primary transform core matrix.

Next, the different types of transform cores (transform core matrices)will be explained. As an example, a 64-point DCT-2 transform core may berepresented using 8-bit integers, where the left half of the even rows(row 0, 2, 4, . . . , 30) of the transform core matrix form a matrixwhich is identical to the 32-point DCT-2 transform core matrix. In oneembodiment, the 64-point DCT-2 core matrix may be derived by setting theparameters as follows: {aa, ab, ac, ad, ae, af, ag, ah, ai, aj, ak, al,am, an, ao, ap, aq, ar, as, at, au, av, aw, ax, ay, az, ba, bb, bc, bd,be, bf, bg, bh, bi, bj, bk, bl, bm, bn, bo, bp, bq, br, bs, bt, bu, by,bw, bx, by, bz, ca, cb, cc, cd, ce, cf, cg, ch, ci, cj, ck}={64, 83, 36,89, 75, 50, 18, 90, 87, 80, 70, 57, 43, 25, 9, 90, 90, 88, 85, 82, 78,73, 67, 61, 54, 46, 38, 31, 22, 13, 4, 91, 90, 90, 90, 88, 87, 86, 84,83, 81, 79, 77, 73, 71, 69, 65, 62, 59, 56, 52, 48, 44, 41, 37, 33, 28,24, 20, 15, 11, 7, 2}.

A 128-point DCT-2 transform core may be represented using 8-bitintegers, and the left half of the even rows (row 0, 2, 4, . . . , 126)of the transform core matrix may form a matrix which is identical to the64-point DCT-2 transform core matrix. In one embodiment, the 128-pointDCT-2 core matrix may be derived by setting the parameters as follows:{aa, ab, ac, ad, ae, af, ag, ah, ai, aj, ak, al, am, an, ao, ap, aq, ar,as, at, au, av, aw, ax, ay, az, ba, bb, bc, bd, be, bf, bg, bh, bi, bj,bk, bl, bm, bn, bo, bp, bq, br, bs, bt, bu, by, bw, bx, by, bz, ca, cb,cc, cd, ce, cf, cg, ch, ci, cj, ck, cl, cm, cn, co, cp, cq, cr, cs, ct,cu, cv, cw, cx, cy, cz, da, db, dc, dd, de, df, dg, dh, di, dj, dk, dl,dm, dn, d_, dp, dq, dr, ds, dt, du, dv, dw, dx, dy, dz, ea, eb, ec, ed,ee, ef, eg, eh, ei, ej, ek, el, em, en, eo, ep, eq, er, es, et, eu, ev,ew}={64, 83, 36, 89, 75, 50, 18, 90, 87, 80, 70, 57, 43, 25, 9, 90, 90,88, 85, 82, 78, 73, 67, 61, 54, 46, 38, 31, 22, 13, 4, 91, 90, 90, 90,88, 87, 86, 84, 83, 81, 79, 77, 73, 71, 69, 65, 62, 59, 56, 52, 48, 44,41, 37, 33, 28, 24, 20, 15, 11, 7, 2, 90, 90, 90, 90, 90, 90, 89, 89,89, 88, 88, 87, 86, 86, 85, 84, 83, 82, 81, 80, 79, 78, 77, 76, 75, 73,72, 71, 69, 68, 66, 65, 63, 62, 60, 58, 58, 55, 53, 51, 49, 48, 45, 44,42, 40, 38, 36, 34, 32, 29, 27, 25, 23, 21, 19, 17, 15, 12, 10, 8, 6, 3,1}.

A B-bit N-point primary transform core, which is composed by M uniquenumbers (despite the signs and zeros), may be derived by the followingsteps: (a) derive an initial transform core T1 by scaling the originaltransform core which is composed by floating-point numbers, i.e., T0,using a scaling factor 2^(B-2)×log₂(N), i.e., T1=T0×2^(B-2)×log₂(N),where the row vectors of T0, i.e., basis vectors, are orthogonal to eachother, and the norm of each row vector is 1, (b) derive an adjustedtransform core T2 by trying adding offsets on a selected set of M uniquenumbers which are used to construct the primary transform core T2, and(c) calculate the cost value C as the sum of absolute values of allelements of (T1*T1^(T)−2^(2B-4)×log₂(N)²×I), where I is an N×N identitymatrix. The offset values which minimize the cost value may be thenadded to T2, and T2 may be updated. Example offset values for each ofthe M unique numbers may include, but are not limited to: −2, −1, 0, +1,+2. After minimizing the cost value by trying all possible combinationof offset values on a selected set of M unique numbers, T2 may be outputas the B-bit N-point primary transform core.

When deriving the N-point DCT-8 and DST-7 transform cores, it may berestricted that the set of M unique numbers which are used to constructthe N-point DCT-8 core matrix is the same set of M unique numbers whichconstruct the N-point DST-7 core matrix. In one embodiment, M equals toN. In one embodiment, the 8-bit DCT-8 and DST-7 transform cores areshown below. For example, in 4-point DST-7 core, there may be 4 uniquenumbers including 28, 55, 74, 84, and this is the same set of 4 uniquenumbers used to construct 4-point DCT-8 core.

4-Point DST-7:

-   -   {29, 55, 74, 84}    -   {74, 74, 0-74}    -   {84, −29, −74, 55}    -   {55, −84, 74, −29}

8-Point DST-7:

-   -   {17, 32, 46, 60, 71, 78, 85, 86}    -   {46, 78, 86, 71, 32, −17, −60, −85}    -   {71, 85, 32, −46, −86, −60, 17, 78}    -   {85, 46, −60, −78, 17, 86, 32, −71}    -   {86, −17, −85, 32, 78, −46, −71, 60}    -   {78, −71, −17, 85, −60, −32, 86, −46}    -   {60, −86, 71, −17, −46, 85, −78, 32}    -   {32, −60, 78, −86, 85, −71, 46, −17}

16-Point DST-7:

-   -   {9, 17, 25, 33, 41, 49, 56, 62, 66, 72, 77, 81, 83, 87, 89, 90}    -   {25, 49, 66, 81, 89, 89, 81, 66, 49, 25, 0, −25, −49, −66, −81,        −89}    -   {41, 72, 89, 83, 62, 25, −17, −56, −81, −90, −77, −49, −9, 33,        66, 87}    -   {56, 87, 81, 41, −17, −66, −90, −72, −25, 33, 77, 89, 62, 9,        −49, −83}    -   {66, 89, 49, −25, −81, −81, −25, 49, 89, 66, 0, −66, −89, −49,        25, 81}    -   {77, 77, 0, −77, −77, 0, 77, 77, 0, −77, −77, 0, 77, 77, 0, −77}    -   {83, 56, −49, −87, −9, 81, 62, −41, −89, −17, 77, 66, −33, −90,        −25, 72}    -   {89, 25, −81, −49, 66, 66, −49, −81, 25, 89, 0, −89, −25, 81,        49, −66}    -   {90, −9, −89, 17, 87, −25, −83, 33, 81, −41, −77, 49, 72, −56,        −66, 62}    -   {87, −41, −66, 72, 33, −89, 9, 83, −49, −62, 77, 25, −90, 17,        81, −56}    -   {81, −66, −25, 89, −49, −49, 89, −25, −66, 81, 0, −81, 66, 25,        −89, 49}    -   {72, −83, 25, 56, −90, 49, 33, −87, 66, 9, −77, 81, −17, −62,        89, −41}    -   {62, −90, 66, −9, −56, 89, −72, 17, 49, −87, 77, −25, −41, 83,        −81, 33}    -   {49, −81, 89, −66, 25, 25, −66, 89, −81, 49, 0, −49, 81, −89,        66, −25}    -   {33, −62, 81, −90, 83, −66, 41, −9, −25, 56, −77, 89, −87, 72,        −49, 17}    -   {17, −33, 49, −62, 72, −81, 87, −90, 89, −83, 77, −66, 56, −41,        25, −9}

32-Point DST-7:

-   -   {4, 9, 13, 17, 21, 26, 30, 34, 38, 42, 45, 50, 53, 56, 60, 63,        66, 68, 72, 74, 77, 78, 80, 82, 84, 85, 86, 88, 88, 89, 90, 90}    -   {13, 26, 38, 50, 60, 68, 77, 82, 86, 89, 90, 88, 85, 80, 74, 66,        56, 45, 34, 21, 9, −4, −17, −30, −42, −53, −63, −72, −78, −84,        −88, −90}    -   {21, 42, 60, 74, 84, 89, 89, 84, 74, 60, 42, 21, 0, −21, −42,        −60, −74, −84, −89, −89, −84, −74, −60, −42, −21, 0, 21, 42, 60,        74, 84, 89}    -   {30, 56, 77, 88, 89, 80, 63, 38, 9, −21, −50, −72, −85, −90,        −84, −68, −45, −17, 13, 42, 66, 82, 90, 86, 74, 53, 26, −4, −34,        −60, −78, −88}    -   {38, 68, 86, 88, 74, 45, 9, −30, −63, −84, −90, −78, −53, −17,        21, 56, 80, 90, 82, 60, 26, −13, −50, −77, −89, −85, −66, −34,        4, 42, 72, 88}    -   {45, 78, 90, 77, 42, −4, −50, −80, −90, −74, −38, 9, 53, 82, 89,        72, 34, −13, −56, −84, −88, −68, −30, 17, 60, 85, 88, 66, 26,        −21, −63, −86}    -   {53, 85, 85, 53, 0, −53, −85, −85, −53, 0, 53, 85, 85, 53, 0,        −53, −85, −85, −53, 0, 53, 85, 85, 53, 0, −53, −85, −85, −53, 0,        53, 85}    -   {60, 89, 74, 21, −42, −84, −84, −42, 21, 74, 89, 60, 0, −60,        −89, −74, −21, 42, 84, 84, 42, −21, −74, −89, −60, 0, 60, 89,        74, 21, −42, −84}    -   {66, 90, 56, −13, −74, −88, −45, 26, 80, 84, 34, −38, −85, −78,        −21, 50, 88, 72, 9, −60, −90, −63, 4, 68, 89, 53, −17, −77, −86,        −42, 30, 82}    -   {72, 86, 34, −45, −89, −63, 13, 78, 82, 21, −56, −90, −53, 26,        84, 77, 9, −66, −88, −42, 38, 88, 68, −4, −74, −85, −30, 50, 90,        60, −17, −80}    -   {77, 80, 9, −72, −84, −17, 66, 86, 26, −60, −88, −34, 53, 90,        42, −45, −90, −50, 38, 89, 56, −30, −88, −63, 21, 85, 68, −13,        −82, −74, 4, 78}    -   {80, 72, −17, −86, −60, 34, 90, 45, −50, −89, −30, 63, 85, 13,        −74, −78, 4, 82, 68, −21, −88, −56, 38, 90, 42, −53, −88, −26,        66, 84, 9, −77}    -   {84, 60, −42, −89, −21, 74, 74, −21, −89, −42, 60, 84, 0, −84,        −60, 42, 89, 21, −74, −74, 21, 89, 42, −60, −84, 0, 84, 60, −42,        −89, −21, 74}    -   {86, 45, −63, −78, 21, 90, 26, −77, −66, 42, 88, 4, −85, −50,        60, 80, −17, −90, −30, 74, 68, −38, −88, −9, 84, 53, −56, −82,        13, 89, 34, −72}    -   {88, 30, −78, −56, 60, 77, −34, −88, 4, 89, 26, −80, −53, 63,        74, −38, −86, 9, 90, 21, −82, −50, 66, 72, −42, −85, 13, 90, 17,        −84, −45, 68}    -   {90, 13, −88, −26, 84, 38, −78, −50, 72, 60, −63, −68, 53, 77,        −42, −82, 30, 86, −17, −89, 4, 90, 9, −88, −21, 85, 34, −80,        −45, 74, 56, −66}    -   {90, −4, −90, 9, 89, −13, −88, 17, 88, −21, −86, 26, 85, −30,        −84, 34, 82, −38, −80, 42, 78, −45, −77, 50, 74, −53, −72, 56,        68, −60, −66, 63}    -   {89, −21, −84, 42, 74, −60, −60, 74, 42, −84, −21, 89, 0, −89,        21, 84, −42, −74, 60, 60, −74, −42, 84, 21, −89, 0, 89, −21,        −84, 42, 74, −60}    -   {88, −38, −72, 68, 42, −86, −4, 88, −34, −74, 66, 45, −85, −9,        89, −30, −77, 63, 50, −84, −13, 90, −26, −78, 60, 53, −82, −17,        90, −21, −80, 56}    -   {85, −53, −53, 85, 0, −85, 53, 53, −85, 0, 85, −53, −53, 85, 0,        −85, 53, 53, −85, 0, 85, −53, −53, 85, 0, −85, 53, 53, −85, 0,        85, −53}    -   {82, −66, −30, 90, −42, −56, 86, −13, −77, 74, 17, −88, 53, 45,        −89, 26, 68, −80, −4, 84, −63, −34, 90, −38, −60, 85, −9, −78,        72, 21, −88, 50}    -   {78, −77, −4, 80, −74, −9, 82, −72, −13, 84, −68, −17, 85, −66,        −21, 86, −63, −26, 88, −60, −30, 88, −56, −34, 89, −53, −38, 90,        −50, −42, 90, −45}    -   {74, −84, 21, 60, −89, 42, 42, −89, 60, 21, −84, 74, 0, −74, 84,        −21, −60, 89, −42, −42, 89, −60, −21, 84, −74, 0, 74, −84, 21,        60, −89, 42}    -   {68, −88, 45, 30, −84, 78, −17, −56, 90, −60, −13, 77, −85, 34,        42, −88, 72, −4, −66, 89, −50, −26, 82, −80, 21, 53, −90, 63, 9,        −74, 86, −38}    -   {63, −90, 66, −4, −60, 90, −68, 9, 56, −89, 72, −13, −53, 88,        −74, 17, 50, −88, 77, −21, −45, 86, −78, 26, 42, −85, 80, −30,        −38, 84, −82, 34}    -   {56, −88, 80, −38, −21, 72, −90, 68, −17, −42, 82, −86, 53, 4,        −60, 88, −78, 34, 26, −74, 90, −66, 13, 45, −84, 85, −50, −9,        63, −89, 77, −30}    -   {50, −82, 88, −66, 21, 30, −72, 90, −78, 42, 9, −56, 85, −86,        60, −13, −38, 77, −90, 74, −34, −17, 63, −88, 84, −53, 4, 45,        −80, 89, −68, 26}    -   {42, −74, 89, −84, 60, −21, −21, 60, −84, 89, −74, 42, 0, −42,        74, −89, 84, −60, 21, 21, −60, 84, −89, 74, −42, 0, 42, −74, 89,        −84, 60, −21}    -   {34, −63, 82, −90, 84, −66, 38, −4, −30, 60, −80, 90, −85, 68,        −42, 9, 26, −56, 78, −89, 86, −72, 45, −13, −21, 53, −77, 88,        −88, 74, −50, 17}    -   {26, −50, 68, −82, 89, −88, 80, −66, 45, −21, −4, 30, −53, 72,        −84, 90, −88, 78, −63, 42, −17, −9, 34, −56, 74, −85, 90, −86,        77, −60, 38, −13}    -   {17, −34, 50, −63, 74, −82, 88, −90, 88, −84, 77, −66, 53, −38,        21, −4, −13, 30, −45, 60, −72, 80, −86, 90, −89, 85, −78, 68,        −56, 42, −26, 9}    -   {9, −17, 26, −34, 42, −50, 56, −63, 68, −74, 78, −82, 85, −88,        89, −90, 90, −88, 86, −84, 80, −77, 72, −66, 60, −53, 45, −38,        30, −21, 13, −4}

64-Point DST-7:

-   -   {2, 5, 7, 8, 11, 13, 15, 17, 20, 22, 23, 26, 29, 30, 32, 34, 37,        38, 41, 42, 44, 46, 48, 49, 52, 53, 55, 58, 59, 60, 62, 64, 65,        66, 68, 69, 72, 72, 74, 73, 76, 77, 78, 79, 81, 80, 82, 83, 85,        84, 85, 86, 87, 87, 88, 89, 89, 90, 89, 90, 90, 90, 90, 90}    -   {7, 13, 20, 26, 32, 38, 44, 49, 55, 60, 65, 69, 74, 77, 81, 83,        85, 87, 89, 90, 90, 90, 90, 89, 87, 85, 83, 81, 77, 74, 69, 65,        60, 55, 49, 44, 38, 32, 26, 20, 13, 7, 0, −7, −13, −20, −26,        −32, −38, −44, −49, −55, −60, −65, −69, −74, −77, −81, −83, −85,        −87, −89, −90, −90}    -   {11, 22, 32, 42, 52, 60, 68, 73, 81, 84, 88, 90, 90, 89, 87, 85,        79, 74, 66, 59, 49, 41, 30, 20, 8, −2, −13, −23, −34, −44, −53,        −62, −69, −76, −80, −85, −89, −90, −90, −90, −87, −83, −78, −72,        −65, −58, −48, −38, −29, −17, −7, 5, 15, 26, 37, 46, 55, 64, 72,        77, 82, 86, 89, 90}    -   {15, 30, 44, 58, 68, 77, 85, 89, 90, 89, 86, 81, 72, 62, 49, 37,        22, 7, −8, −23, −38, −52, −64, −74, −80, −87, −90, −90, −88,        −83, −76, −66, −55, −42, −29, −13, 2, 17, 32, 46, 59, 69, 78,        84, 89, 90, 90, 85, 79, 72, 60, 48, 34, 20, 5, −11, −26, −41,        −53, −65, −73, −82, −87, −90}    -   {20, 38, 55, 69, 81, 87, 90, 89, 83, 74, 60, 44, 26, 7, −13,        −32, −49, −65, −77, −85, −90, −90, −85, −77, −65, −49, −32, −13,        7, 26, 44, 60, 74, 83, 89, 90, 87, 81, 69, 55, 38, 20, 0, −20,        −38, −55, −69, −81, −87, −90, −89, −83, −74, −60, −44, −26, −7,        13, 32, 49, 65, 77, 85, 90}    -   {23, 46, 65, 79, 88, 90, 86, 76, 60, 41, 17, −7, −30, −52, −69,        −82, −90, −90, −85, −72, −55, −34, −11, 13, 37, 58, 74, 84, 90,        89, 80, 68, 49, 29, 5, −20, −42, −62, −77, −87, −90, −87, −78,        −64, −44, −22, 2, 26, 48, 66, 81, 89, 90, 85, 73, 59, 38, 15,        −8, −32, −53, −72, −83, −89}    -   {29, 53, 74, 86, 90, 85, 72, 52, 26, −2, −30, −55, −73, −87,        −90, −84, −72, −49, −23, 5, 32, 58, 76, 87, 90, 85, 69, 48, 22,        −7, −34, −59, −77, −88, −90, −83, −68, −46, −20, 8, 37, 60, 78,        89, 90, 82, 66, 44, 17, −11, −38, −62, −79, −89, −89, −80, −65,        −42, −15, 13, 41, 64, 81, 90}    -   {32, 60, 81, 90, 87, 74, 49, 20, −13, −44, −69, −85, −90, −83,        −65, −38, −7, 26, 55, 77, 89, 89, 77, 55, 26, −7, −38, −65, −83,        −90, −85, −69, −44, −13, 20, 49, 74, 87, 90, 81, 60, 32, 0, −32,        −60, −81, −90, −87, −74, −49, −20, 13, 44, 69, 85, 90, 83, 65,        38, 7, −26, −55, −77, −89}    -   {37, 66, 85, 90, 79, 55, 22, −15, −49, −76, −90, −87, −72, −42,        −7, 30, 62, 83, 90, 82, 60, 29, −8, −44, −72, −88, −89, −73,        −48, −13, 23, 58, 81, 90, 84, 65, 34, −2, −38, −68, −86, −90,        −78, −53, −20, 17, 52, 77, 89, 87, 69, 41, 5, −32, −64, −85,        −90, −80, −59, −26, 11, 46, 74, 89}    -   {41, 72, 89, 87, 66, 32, −8, −48, −77, −90, −85, −60, −23, 17,        55, 80, 90, 81, 53, 15, −26, −62, −84, −90, −76, −46, −7, 34,        68, 87, 89, 72, 38, −2, −42, −74, −90, −86, −65, −30, 11, 49,        78, 90, 83, 59, 22, −20, −58, −82, −90, −79, −52, −13, 29, 64,        85, 89, 73, 44, 5, −37, −69, −88}    -   {44, 77, 90, 81, 49, 7, −38, −74, −90, −83, −55, −13, 32, 69,        89, 85, 60, 20, −26, −65, −87, −87, −65, −26, 20, 60, 85, 89,        69, 32, −13, −55, −83, −90, −74, −38, 7, 49, 81, 90, 77, 44, 0,        −44, −77, −90, −81, −49, −7, 38, 74, 90, 83, 55, 13, −32, −69,        −89, −85, −60, −20, 26, 65, 87}    -   {48, 80, 90, 72, 30, −20, −64, −88, −85, −58, −11, 38, 76, 90,        77, 41, −8, −55, −84, −89, −65, −22, 29, 69, 89, 82, 49, 2, −46,        −81, −90, −72, −32, 17, 62, 87, 86, 59, 13, −37, −73, −90, −78,        −42, 7, 53, 85, 89, 66, 23, −26, −68, −90, −83, −52, −5, 44, 79,        90, 74, 34, −15, −60, −87}    -   {52, 84, 87, 59, 8, −44, −80, −90, −65, −17, 37, 77, 90, 72, 26,        −29, −72, −90, −76, −34, 20, 66, 89, 81, 42, −11, −60, −88, −85,        −49, 2, 53, 85, 87, 58, 7, −46, −82, −89, −64, −15, 38, 78, 90,        69, 23, −30, −74, −90, −73, −32, 22, 68, 90, 79, 41, −13, −62,        −89, −83, −48, 5, 55, 86}    -   {55, 87, 83, 44, −13, −65, −90, −77, −32, 26, 74, 90, 69, 20,        −38, −81, −89, −60, −7, 49, 85, 85, 49, −7, −60, −89, −81, −38,        20, 69, 90, 74, 26, −32, −77, −90, −65, −13, 44, 83, 87, 55, 0,        −55, −87, −83, −44, 13, 65, 90, 77, 32, −26, −74, −90, −69, −20,        38, 81, 89, 60, 7, −49, −85}    -   {59, 90, 77, 29, −34, −81, −88, −53, 7, 64, 90, 74, 22, −41,        −83, −86, −48, 13, 68, 90, 69, 15, −46, −85, −85, −42, 20, 72,        90, 65, 8, −52, −87, −80, −37, 26, 76, 89, 60, 2, −58, −89, −78,        −30, 32, 79, 89, 55, −5, −62, −90, −73, −23, 38, 82, 87, 49,        −11, −66, −90, −72, −17, 44, 84}    -   {62, 90, 69, 11, −53, −89, −76, −22, 44, 86, 80, 32, −34, −82,        −85, −42, 23, 77, 89, 52, −13, −72, −90, −60, 2, 64, 90, 68, 8,        −55, −90, −73, −20, 46, 87, 81, 30, −37, −83, −84, −41, 26, 78,        88, 49, −15, −72, −90, −59, 5, 65, 90, 66, 7, −58, −89, −74,        −17, 48, 87, 79, 29, −38, −85}    -   {65, 90, 60, −7, −69, −90, −55, 13, 74, 89, 49, −20, −77, −87,        −44, 26, 81, 85, 38, −32, −83, −83, −32, 38, 85, 81, 26, −44,        −87, −77, −20, 49, 89, 74, 13, −55, −90, −69, −7, 60, 90, 65, 0,        −65, −90, −60, 7, 69, 90, 55, −13, −74, −89, −49, 20, 77, 87,        44, −26, −81, −85, −38, 32, 83}    -   {68, 89, 49, −23, −80, −83, −29, 46, 89, 72, 5, −65, −90, −53,        20, 79, 84, 32, −42, −88, −74, −8, 62, 90, 58, −15, −77, −86,        −37, 38, 87, 76, 13, −59, −90, −60, 11, 73, 87, 41, −34, −85,        −78, −17, 55, 90, 64, −7, −72, −89, −44, 30, 85, 81, 22, −52,        −90, −66, 2, 69, 90, 48, −26, −82}    -   {72, 88, 38, −41, −89, −69, 2, 72, 87, 37, −42, −89, −68, 5, 74,        87, 34, −44, −90, −66, 7, 73, 86, 32, −46, −89, −65, 8, 76, 85,        30, −48, −90, −64, 11, 77, 84, 29, −49, −90, −62, 13, 78, 85,        26, −52, −90, −60, 15, 79, 83, 23, −53, −90, −59, 17, 81, 82,        22, −55, −90, −58, 20, 80}    -   {74, 85, 26, −55, −90, −49, 32, 87, 69, −7, −77, −83, −20, 60,        90, 44, −38, −89, −65, 13, 81, 81, 13, −65, −89, −38, 44, 90,        60, −20, −83, −77, −7, 69, 87, 32, −49, −90, −55, 26, 85, 74, 0,        −74, −85, −26, 55, 90, 49, −32, −87, −69, 7, 77, 83, 20, −60,        −90, −44, 38, 89, 65, −13, −81}    -   {76, 82, 13, −68, −87, −26, 59, 89, 38, −48, −90, −49, 37, 90,        60, −23, −86, −69, 11, 80, 77, 2, −73, −83, −15, 66, 87, 29,        −58, −90, −41, 46, 90, 52, −34, −89, −62, 22, 85, 72, −8, −81,        −78, −5, 74, 85, 17, −65, −88, −30, 55, 90, 42, −44, −90, −53,        32, 89, 64, −20, −84, −72, 7, 79}    -   {78, 78, 0, −78, −78, 0, 78, 78, 0, −78, −78, 0, 78, 78, 0, −78,        −78, 0, 78, 78, 0, −78, −78, 0, 78, 78, 0, −78, −78, 0, 78, 78,        0, −78, −78, 0, 78, 78, 0, −78, −78, 0, 78, 78, 0, −78, −78, 0,        78, 78, 0, −78, −78, 0, 78, 78, 0, −78, −78, 0, 78, 78, 0, −78}    -   {81, 74, −13, −85, −65, 26, 89, 55, −38, −90, −44, 49, 90, 32,        −60, −87, −20, 69, 83, 7, −77, −77, 7, 83, 69, −20, −87, −60,        32, 90, 49, −44, −90, −38, 55, 89, 26, −65, −85, −13, 74, 81, 0,        −81, −74, 13, 85, 65, −26, −89, −55, 38, 90, 44, −49, −90, −32,        60, 87, 20, −69, −83, −7, 77}    -   {82, 68, −26, −89, −48, 49, 90, 23, −69, −80, 2, 83, 66, −29,        −90, −46, 52, 89, 22, −72, −81, 5, 85, 65, −30, −90, −44, 53,        89, 20, −72, −79, 7, 84, 64, −32, −90, −42, 55, 88, 17, −74,        −78, 8, 85, 62, −34, −90, −41, 58, 87, 15, −73, −77, 11, 86, 60,        −37, −90, −38, 59, 87, 13, −76}    -   {85, 62, −38, −90, −29, 69, 79, −11, −87, −53, 48, 89, 17, −76,        −74, 22, 89, 44, −58, −86, −7, 80, 66, −32, −90, −34, 65, 82,        −5, −85, −59, 42, 90, 23, −72, −77, 15, 89, 49, −52, −88, −13,        78, 72, −26, −90, −41, 60, 84, 2, −83, −64, 37, 90, 30, −68,        −81, 8, 87, 55, −46, −90, −20, 73}    -   {85, 55, −49, −87, −7, 83, 60, −44, −89, −13, 81, 65, −38, −90,        −20, 77, 69, −32, −90, −26, 74, 74, −26, −90, −32, 69, 77, −20,        −90, −38, 65, 81, −13, −89, −44, 60, 83, −7, −87, −49, 55, 85,        0, −85, −55, 49, 87, 7, −83, −60, 44, 89, 13, −81, −65, 38, 90,        20, −77, −69, 32, 90, 26, −74}    -   {87, 48, −60, −80, 15, 90, 34, −72, −74, 30, 90, 20, −79, −64,        44, 88, 5, −85, −52, 58, 83, −11, −90, −38, 68, 76, −26, −90,        −23, 77, 66, −41, −89, −8, 85, 55, −53, −84, 7, 89, 42, −65,        −78, 22, 90, 29, −73, −69, 37, 89, 13, −82, −59, 49, 86, −2,        −87, −46, 62, 81, −17, −90, −32, 72}    -   {88, 41, −69, −72, 37, 89, 5, −87, −44, 66, 73, −32, −89, −8,        85, 48, −64, −77, 29, 90, 13, −85, −52, 60, 79, −23, −90, −17,        82, 55, −58, −80, 20, 90, 22, −81, −59, 53, 83, −15, −90, −26,        78, 62, −49, −84, 11, 90, 30, −76, −65, 46, 86, −7, −90, −34,        74, 68, −42, −87, 2, 89, 38, −72}    -   {89, 32, −77, −60, 55, 81, −26, −90, −7, 87, 38, −74, −65, 49,        83, −20, −90, −13, 85, 44, −69, −69, 44, 85, −13, −90, −20, 83,        49, −65, −74, 38, 87, −7, −90, −26, 81, 55, −60, −77, 32, 89, 0,        −89, −32, 77, 60, −55, −81, 26, 90, 7, −87, −38, 74, 65, −49,        −83, 20, 90, 13, −85, −44, 69}    -   {89, 23, −83, −46, 72, 65, −53, −79, 32, 88, −8, −90, −15, 86,        38, −76, −59, 60, 73, −41, −85, 17, 90, 7, −89, −30, 81, 52,        −66, −69, 48, 82, −26, −90, 2, 90, 22, −85, −44, 72, 64, −55,        −78, 34, 87, −11, −90, −13, 87, 37, −77, −58, 62, 74, −42, −84,        20, 90, 5, −89, −29, 80, 49, −68}    -   {90, 15, −87, −30, 82, 44, −73, −58, 65, 68, −53, −77, 41, 85,        −26, −89, 11, 90, 5, −89, −20, 86, 34, −81, −48, 72, 60, −62,        −72, 49, 79, −37, −85, 22, 90, −7, −90, −8, 89, 23, −84, −38,        78, 52, −69, −64, 59, 74, −46, −80, 32, 87, −17, −90, 2, 90, 13,        −88, −29, 83, 42, −76, −55, 66}    -   {90, 7, −90, −13, 89, 20, −87, −26, 85, 32, −83, −38, 81, 44,        −77, −49, 74, 55, −69, −60, 65, 65, −60, −69, 55, 74, −49, −77,        44, 81, −38, −83, 32, 85, −26, −87, 20, 89, −13, −90, 7, 90, 0,        −90, −7, 90, 13, −89, −20, 87, 26, −85, −32, 83, 38, −81, −44,        77, 49, −74, −55, 69, 60, −65}    -   {90, −2, −90, 5, 90, −7, −90, 8, 90, −11, −89, 13, 90, −15, −89,        17, 89, −20, −88, 22, 87, −23, −87, 26, 86, −29, −85, 30, 84,        −32, −85, 34, 83, −37, −82, 38, 80, −41, −81, 42, 79, −44, −78,        46, 77, −48, −76, 49, 73, −52, −74, 53, 72, −55, −72, 58, 69,        −59, −68, 60, 66, −62, −65, 64}    -   {90, −11, −89, 22, 86, −32, −82, 42, 77, −52, −72, 60, 64, −68,        −55, 73, 46, −81, −37, 84, 26, −88, −15, 90, 5, −90, 7, 89, −17,        −87, 29, 85, −38, −79, 48, 74, −58, −66, 65, 59, −72, −49, 78,        41, −83, −30, 87, 20, −90, −8, 90, −2, −90, 13, 89, −23, −85,        34, 80, −44, −76, 53, 69, −62}    -   {90, −20, −85, 38, 77, −55, −65, 69, 49, −81, −32, 87, 13, −90,        7, 89, −26, −83, 44, 74, −60, −60, 74, 44, −83, −26, 89, 7, −90,        13, 87, −32, −81, 49, 69, −65, −55, 77, 38, −85, −20, 90, 0,        −90, 20, 85, −38, −77, 55, 65, −69, −49, 81, 32, −87, −13, 90,        −7, −89, 26, 83, −44, −74, 60}    -   {90, −29, −81, 53, 64, −74, −41, 86, 13, −90, 15, 85, −42, −72,        65, 52, −80, −26, 89, −2, −89, 30, 79, −55, −62, 73, 38, −87,        −11, 90, −17, −84, 44, 72, −66, −49, 82, 23, −90, 5, 89, −32,        −78, 58, 60, −76, −37, 87, 8, −90, 20, 85, −46, −69, 68, 48,        −83, −22, 90, −7, −88, 34, 77, −59}    -   {89, −37, −74, 66, 46, −85, −11, 90, −26, −79, 59, 55, −80, −22,        90, −15, −85, 49, 64, −76, −32, 90, −5, −87, 41, 72, −69, −42,        87, 7, −89, 30, 77, −62, −52, 83, 17, −90, 20, 82, −53, −60, 78,        29, −90, 8, 86, −44, −68, 72, 38, −88, −2, 89, −34, −73, 65, 48,        −84, −13, 90, −23, −81, 58}    -   {87, −44, −65, 77, 26, −90, 20, 81, −60, −49, 85, 7, −89, 38,        69, −74, −32, 90, −13, −83, 55, 55, −83, −13, 90, −32, −74, 69,        38, −89, 7, 85, −49, −60, 81, 20, −90, 26, 77, −65, −44, 87, 0,        −87, 44, 65, −77, −26, 90, −20, −81, 60, 49, −85, −7, 89, −38,        −69, 74, 32, −90, 13, 83, −55}    -   {86, −52, −55, 84, 5, −87, 48, 59, −83, −8, 89, −44, −62, 80,        13, −90, 41, 65, −79, −17, 90, −37, −68, 77, 22, −90, 32, 72,        −73, −26, 90, −29, −74, 72, 30, −90, 23, 76, −69, −34, 90, −20,        −78, 66, 38, −89, 15, 81, −64, −42, 89, −11, −82, 60, 46, −88,        7, 85, −58, −49, 87, −2, −85, 53}    -   {84, −59, −44, 90, −17, −77, 72, 29, −90, 34, 66, −81, −11, 88,        −49, −53, 87, −7, −82, 64, 38, −90, 23, 74, −73, −22, 90, −41,        −62, 83, 5, −86, 55, 48, −89, 13, 79, −68, −32, 90, −30, −69,        78, 15, −89, 46, 58, −85, 2, 85, −60, −42, 89, −20, −76, 72, 26,        −90, 37, 65, −80, −8, 87, −52}    -   {83, −65, −32, 90, −38, −60, 85, −7, −81, 69, 26, −90, 44, 55,        −87, 13, 77, −74, −20, 89, −49, −49, 89, −20, −74, 77, 13, −87,        55, 44, −90, 26, 69, −81, −7, 85, −60, −38, 90, −32, −65, 83, 0,        −83, 65, 32, −90, 38, 60, −85, 7, 81, −69, −26, 90, −44, −55,        87, −13, −77, 74, 20, −89, 49}    -   {80, −72, −20, 88, −58, −38, 90, −41, −55, 89, −22, −69, 82, −2,        −81, 72, 17, −87, 59, 37, −90, 42, 53, −89, 23, 68, −83, 5, 79,        −74, −15, 87, −60, −34, 90, −44, −52, 90, −26, −66, 85, −7, −78,        73, 13, −86, 62, 32, −90, 46, 49, −89, 29, 65, −84, 8, 77, −76,        −11, 85, −64, −30, 90, −48}    -   {79, −76, −7, 82, −72, −13, 84, −68, −20, 87, −64, −26, 89, −59,        −32, 89, −53, −38, 90, −48, −44, 90, −42, −49, 90, −37, −55, 90,        −30, −60, 88, −23, −65, 86, −17, −69, 85, −11, −74, 80, −5, −77,        78, 2, −81, 73, 8, −83, 72, 15, −85, 66, 22, −87, 62, 29, −89,        58, 34, −90, 52, 41, −90, 46}    -   {77, −81, 7, 74, −83, 13, 69, −85, 20, 65, −87, 26, 60, −89, 32,        55, −90, 38, 49, −90, 44, 44, −90, 49, 38, −90, 55, 32, −89, 60,        26, −87, 65, 20, −85, 69, 13, −83, 74, 7, −81, 77, 0, −77, 81,        −7, −74, 83, −13, −69, 85, −20, −65, 87, −26, −60, 89, −32, −55,        90, −38, −49, 90, −44}    -   {73, −85, 20, 62, −90, 38, 46, −90, 55, 29, −87, 69, 8, −79, 81,        −11, −68, 87, −30, −53, 90, −48, −37, 89, −64, −17, 83, −76, 2,        74, −84, 22, 60, −89, 41, 44, −90, 58, 26, −86, 72, 7, −78, 80,        −13, −66, 88, −32, −52, 90, −49, −34, 89, −65, −15, 82, −77, 5,        72, −85, 23, 59, −90, 42}    -   {72, −87, 32, 48, −90, 60, 17, −80, 81, −15, −62, 90, −46, −34,        87, −72, −2, 74, −86, 30, 49, −90, 59, 20, −82, 79, −13, −64,        89, −44, −37, 88, −69, −5, 73, −85, 29, 52, −90, 58, 22, −83,        78, −11, −65, 90, −42, −38, 89, −68, −7, 76, −84, 26, 53, −90,        55, 23, −85, 77, −8, −66, 89, −41}    -   {69, −89, 44, 32, −85, 77, −13, −60, 90, −55, −20, 81, −83, 26,        49, −90, 65, 7, −74, 87, −38, −38, 87, −74, 7, 65, −90, 49, 26,        −83, 81, −20, −55, 90, −60, −13, 77, −85, 32, 44, −89, 69, 0,        −69, 89, −44, −32, 85, −77, 13, 60, −90, 55, 20, −81, 83, −26,        −49, 90, −65, −7, 74, −87, 38}    -   {66, −90, 55, 15, −76, 87, −42, −30, 83, −82, 29, 44, −88, 73,        −13, −58, 90, −65, −2, 68, −90, 53, 17, −77, 87, −41, −32, 85,        −80, 26, 46, −89, 74, −11, −59, 90, −64, −5, 69, −89, 52, 20,        −78, 86, −38, −34, 84, −81, 23, 48, −89, 72, −8, −60, 90, −62,        −7, 72, −90, 49, 22, −79, 85, −37}    -   {64, −90, 65, −2, −62, 90, −66, 5, 60, −90, 68, −7, −59, 90,        −69, 8, 58, −90, 72, −11, −55, 89, −72, 13, 53, −90, 74, −15,        −52, 89, −73, 17, 49, −89, 76, −20, −48, 88, −77, 22, 46, −87,        78, −23, −44, 87, −79, 26, 42, −86, 81, −29, −41, 85, −80, 30,        38, −84, 82, −32, −37, 85, −83, 34}    -   {60, −90, 74, −20, −44, 85, −83, 38, 26, −77, 89, −55, −7, 65,        −90, 69, −13, −49, 87, −81, 32, 32, −81, 87, −49, −13, 69, −90,        65, −7, −55, 89, −77, 26, 38, −83, 85, −44, −20, 74, −90, 60, 0,        −60, 90, −74, 20, 44, −85, 83, −38, −26, 77, −89, 55, 7, −65,        90, −69, 13, 49, −87, 81, −32}    -   {58, −89, 81, −37, −23, 74, −90, 66, −13, −46, 84, −85, 48, 11,        −65, 90, −73, 26, 34, −79, 89, −59, 2, 55, −88, 80, −38, −22,        72, −90, 68, −15, −44, 85, −86, 49, 8, −64, 90, −76, 29, 32,        −78, 90, −60, 5, 53, −87, 82, −41, −20, 72, −90, 69, −17, −42,        83, −87, 52, 7, −62, 89, −77, 30}    -   {53, −86, 85, −52, −2, 55, −87, 84, −49, −5, 58, −87, 85, −48,        −7, 59, −88, 83, −46, −8, 60, −89, 82, −44, −11, 62, −89, 80,        −42, −13, 64, −90, 81, −41, −15, 65, −89, 79, −38, −17, 66, −90,        78, −37, −20, 68, −90, 77, −34, −22, 69, −90, 76, −32, −23, 72,        −90, 73, −30, −26, 72, −90, 74, −29}    -   {49, −83, 89, −65, 20, 32, −74, 90, −77, 38, 13, −60, 87, −85,        55, −7, −44, 81, −90, 69, −26, −26, 69, −90, 81, −44, −7, 55,        −85, 87, −60, 13, 38, −77, 90, −74, 32, 20, −65, 89, −83, 49, 0,        −49, 83, −89, 65, −20, −32, 74, −90, 77, −38, −13, 60, −87, 85,        −55, 7, 44, −81, 90, −69, 26}    -   {46, −79, 90, −76, 41, 7, −52, 82, −90, 72, −34, −13, 58, −84,        89, −68, 29, 20, −62, 87, −87, 64, −22, −26, 66, −89, 85, −59,        15, 32, −72, 89, −83, 53, −8, −38, 73, −90, 81, −48, 2, 44, −78,        90, −77, 42, 5, −49, 80, −90, 74, −37, −11, 55, −85, 90, −69,        30, 17, −60, 86, −88, 65, −23}    -   {42, −73, 90, −85, 59, −20, −23, 62, −85, 90, −72, 38, 5, −46,        77, −90, 82, −55, 15, 29, −65, 87, −89, 69, −34, −8, 49, −79,        90, −81, 52, −11, −32, 68, −88, 87, −66, 30, 13, −53, 80, −90,        78, −48, 7, 37, −72, 89, −86, 64, −26, −17, 58, −83, 90, −76,        44, −2, −41, 74, −89, 84, −60, 22}    -   {38, −69, 87, −89, 74, −44, 7, 32, −65, 85, −90, 77, −49, 13,        26, −60, 83, −90, 81, −55, 20, 20, −55, 81, −90, 83, −60, 26,        13, −49, 77, −90, 85, −65, 32, 7, −44, 74, −89, 87, −69, 38, 0,        −38, 69, −87, 89, −74, 44, −7, −32, 65, −85, 90, −77, 49, −13,        −26, 60, −83, 90, −81, 55, −20}    -   {34, −64, 83, −90, 85, −65, 37, −2, −32, 62, −82, 90, −84, 66,        −38, 5, 30, −60, 80, −90, 85, −68, 41, −7, −29, 59, −81, 90,        −86, 69, −42, 8, 26, −58, 79, −90, 87, −72, 44, −11, −23, 55,        −78, 89, −87, 72, −46, 13, 22, −53, 77, −90, 88, −74, 48, −15,        −20, 52, −76, 89, −89, 73, −49, 17}    -   {30, −58, 77, −89, 89, −81, 62, −37, 7, 23, −52, 74, −87, 90,        −83, 66, −42, 13, 17, −46, 69, −84, 90, −85, 72, −48, 20, 11,        −41, 65, −82, 90, −87, 73, −53, 26, 5, −34, 60, −79, 90, −89,        78, −59, 32, −2, −29, 55, −76, 88, −90, 80, −64, 38, −8, −22,        49, −72, 86, −90, 85, −68, 44, −15}    -   {26, −49, 69, −83, 90, −89, 81, −65, 44, −20, −7, 32, −55, 74,        −85, 90, −87, 77, −60, 38, −13, −13, 38, −60, 77, −87, 90, −85,        74, −55, 32, −7, −20, 44, −65, 81, −89, 90, −83, 69, −49, 26, 0,        −26, 49, −69, 83, −90, 89, −81, 65, −44, 20, 7, −32, 55, −74,        85, −90, 87, −77, 60, −38, 13}    -   {22, −42, 60, −73, 84, −90, 89, −85, 74, −59, 41, −20, −2, 23,        −44, 62, −76, 85, −90, 90, −83, 72, −58, 38, −17, −5, 26, −46,        64, −77, 86, −90, 89, −82, 72, −55, 37, −15, −7, 29, −48, 65,        −78, 87, −90, 89, −80, 69, −53, 34, −13, −8, 30, −49, 66, −79,        87, −90, 88, −81, 68, −52, 32, −11}    -   {17, −34, 49, −64, 73, −83, 89, −90, 89, −85, 76, −65, 52, −37,        20, −2, −15, 32, −48, 62, −74, 82, −88, 90, −90, 84, −77, 66,        −53, 38, −22, 5, 13, −30, 46, −60, 72, −80, 87, −90, 89, −85,        78, −68, 55, −41, 23, −7, −11, 29, −44, 59, −72, 81, −87, 90,        −90, 86, −79, 69, −58, 42, −26, 8}    -   {13, −26, 38, −49, 60, −69, 77, −83, 87, −90, 90, −89, 85, −81,        74, −65, 55, −44, 32, −20, 7, 7, −20, 32, −44, 55, −65, 74, −81,        85, −89, 90, −90, 87, −83, 77, −69, 60, −49, 38, −26, 13, 0,        −13, 26, −38, 49, −60, 69, −77, 83, −87, 90, −90, 89, −85, 81,        −74, 65, −55, 44, −32, 20, −7}    -   {8, −17, 26, −34, 42, −49, 58, −64, 69, −73, 79, −83, 86, −89,        90, −90, 90, −89, 87, −85, 81, −76, 72, −65, 59, −52, 44, −37,        29, −20, 11, −2, −7, 15, −23, 32, −41, 48, −55, 62, −68, 74,        −78, 82, −85, 88, −89, 90, −90, 90, −87, 84, −80, 77, −72, 66,        −60, 53, −46, 38, −30, 22, −13, 5}    -   {5, −8, 13, −17, 22, −26, 30, −34, 38, −42, 46, −49, 53, −58,        60, −64, 66, −69, 72, −73, 77, −79, 80, −83, 84, −86, 87, −89,        90, −90, 90, −90, 90, −90, 89, −89, 88, −87, 85, −85, 82, −81,        78, −76, 74, −72, 68, −65, 62, −59, 55, −52, 48, −44, 41, −37,        32, −29, 23, −20, 15, −11, 7, −2}

4-Point DCT-8:

-   -   {84, 74, 55, 29}    -   {74, 0, −74, −74}    -   {55, −74, −29, 84}    -   {29, −74, 84, −55}

8-Point DCT-8:

-   -   {86, 85, 78, 71, 60, 46, 32, 17}    -   {85, 60, 17, −32, −71, −86, −78, −46}    -   {78, 17, −60, −86, −46, 32, 85, 71}    -   {71, −32, −86, −17, 78, 60, −46, −85}    -   {60, −71, −46, 78, 32, −85, −17, 86}    -   {46, −86, 32, 60, −85, 17, 71, −78}    -   {32, −78, 85, −46, −17, 71, −86, 60}    -   {17, −46, 71, −85, 86, −78, 60, −32}

16-Point DCT-8:

-   -   {90, 89, 87, 83, 81, 77, 72, 66, 62, 56, 49, 41, 33, 25, 17, 9}    -   {89, 81, 66, 49, 25, 0, −25, −49, −66, −81, −89, −89, −81, −66,        −49, −25}    -   {87, 66, 33, −9, −49, −77, −90, −81, −56, −17, 25, 62, 83, 89,        72, 41}    -   {83, 49, −9, −62, −89, −77, −33, 25, 72, 90, 66, 17, −41, −81,        −87, −56}    -   {81, 25, −49, −89, −66, 0, 66, 89, 49, −25, −81, −81, −25, 49,        89, 66}    -   {77, 0, −77, −77, 0, 77, 77, 0, −77, −77, 0, 77, 77, 0, −77,        −77}    -   {72, −25, −90, −33, 66, 77, −17, −89, −41, 62, 81, −9, −87, −49,        56, 83}    -   {66, −49, −81, 25, 89, 0, −89, −25, 81, 49, −66, −66, 49, 81,        −25, −89}    -   {62, −66, −56, 72, 49, −77, −41, 81, 33, −83, −25, 87, 17, −89,        −9, 90}    -   {56, −81, −17, 90, −25, −77, 62, 49, −83, −9, 89, −33, −72, 66,        41, −87}    -   {49, −89, 25, 66, −81, 0, 81, −66, −25, 89, −49, −49, 89, −25,        −66, 81}    -   {41, −89, 62, 17, −81, 77, −9, −66, 87, −33, −49, 90, −56, −25,        83, −72}    -   {33, −81, 83, −41, −25, 77, −87, 49, 17, −72, 89, −56, −9, 66,        −90, 62}    -   {25, −66, 89, −81, 49, 0, −49, 81, −89, 66, −25, −25, 66, −89,        81, −49}    -   {17, −49, 72, −87, 89, −77, 56, −25, −9, 41, −66, 83, −90, 81,        −62, 33}    -   {9, −25, 41, −56, 66, −77, 83, −89, 90, −87, 81, −72, 62, −49,        33, −17}

32-Point DCT-8:

-   -   {90, 90, 89, 88, 88, 86, 85, 84, 82, 80, 78, 77, 74, 72, 68, 66,        63, 60, 56, 53, 50, 45, 42, 38, 34, 30, 26, 21, 17, 13, 9, 4}    -   {90, 88, 84, 78, 72, 63, 53, 42, 30, 17, 4, −9, −21, −34, −45,        −56, −66, −74, −80, −85, −88, −90, −89, −86, −82, −77, −68, −60,        −50, −38, −26, −13}    -   {89, 84, 74, 60, 42, 21, 0, −21, −42, −60, −74, −84, −89, −89,        −84, −74, −60, −42, −21, 0, 21, 42, 60, 74, 84, 89, 89, 84, 74,        60, 42, 21}    -   {88, 78, 60, 34, 4, −26, −53, −74, −86, −90, −82, −66, −42, −13,        17, 45, 68, 84, 90, 85, 72, 50, 21, −9, −38, −63, −80, −89, −88,        −77, −56, −30}    -   {88, 72, 42, 4, −34, −66, −85, −89, −77, −50, −13, 26, 60, 82,        90, 80, 56, 21, −17, −53, −78, −90, −84, −63, −30, 9, 45, 74,        88, 86, 68, 38}    -   {86, 63, 21, −26, −66, −88, −85, −60, −17, 30, 68, 88, 84, 56,        13, −34, −72, −89, −82, −53, −9, 38, 74, 90, 80, 50, 4, −42,        −77, −90, −78, −45}    -   {85, 53, 0, −53, −85, −85, −53, 0, 53, 85, 85, 53, 0, −53, −85,        −85, −53, 0, 53, 85, 85, 53, 0, −53, −85, −85, −53, 0, 53, 85,        85, 53}    -   {84, 42, −21, −74, −89, −60, 0, 60, 89, 74, 21, −42, −84, −84,        −42, 21, 74, 89, 60, 0, −60, −89, −74, −21, 42, 84, 84, 42, −21,        −74, −89, −60}    -   {82, 30, −42, −86, −77, −17, 53, 89, 68, 4, −63, −90, −60, 9,        72, 88, 50, −21, −78, −85, −38, 34, 84, 80, 26, −45, −88, −74,        −13, 56, 90, 66}    -   {80, 17, −60, −90, −50, 30, 85, 74, 4, −68, −88, −38, 42, 88,        66, −9, −77, −84, −26, 53, 90, 56, −21, −82, −78, −13, 63, 89,        45, −34, −86, −72}    -   {78, 4, −74, −82, −13, 68, 85, 21, −63, −88, −30, 56, 89, 38,        −50, −90, −45, 42, 90, 53, −34, −88, −60, 26, 86, 66, −17, −84,        −72, 9, 80, 77}    -   {77, −9, −84, −66, 26, 88, 53, −42, −90, −38, 56, 88, 21, −68,        −82, −4, 78, 74, −13, −85, −63, 30, 89, 50, −45, −90, −34, 60,        86, 17, −72, −80}    -   {74, −21, −89, −42, 60, 84, 0, −84, −60, 42, 89, 21, −74, −74,        21, 89, 42, −60, −84, 0, 84, 60, −42, −89, −21, 74, 74, −21,        −89, −42, 60, 84}    -   {72, −34, −89, −13, 82, 56, −53, −84, 9, 88, 38, −68, −74, 30,        90, 17, −80, −60, 50, 85, −4, −88, −42, 66, 77, −26, −90, −21,        78, 63, −45, −86}    -   {68, −45, −84, 17, 90, 13, −85, −42, 72, 66, −50, −82, 21, 90,        9, −86, −38, 74, 63, −53, −80, 26, 89, 4, −88, −34, 77, 60, −56,        −78, 30, 88}    -   {66, −56, −74, 45, 80, −34, −85, 21, 88, −9, −90, −4, 89, 17,        −86, −30, 82, 42, −77, −53, 68, 63, −60, −72, 50, 78, −38, −84,        26, 88, −13, −90}    -   {63, −66, −60, 68, 56, −72, −53, 74, 50, −77, −45, 78, 42, −80,        −38, 82, 34, −84, −30, 85, 26, −86, −21, 88, 17, −88, −13, 89,        9, −90, −4, 90}    -   {60, −74, −42, 84, 21, −89, 0, 89, −21, −84, 42, 74, −60, −60,        74, 42, −84, −21, 89, 0, −89, 21, 84, −42, −74, 60, 60, −74,        −42, 84, 21, −89}    -   {56, −80, −21, 90, −17, −82, 53, 60, −78, −26, 90, −13, −84, 50,        63, −77, −30, 89, −9, −85, 45, 66, −74, −34, 88, −4, −86, 42,        68, −72, −38, 88}    -   {53, −85, 0, 85, −53, −53, 85, 0, −85, 53, 53, −85, 0, 85, −53,        −53, 85, 0, −85, 53, 53, −85, 0, 85, −53, −53, 85, 0, −85, 53,        53, −85}    -   {50, −88, 21, 72, −78, −9, 85, −60, −38, 90, −34, −63, 84, −4,        −80, 68, 26, −89, 45, 53, −88, 17, 74, −77, −13, 86, −56, −42,        90, −30, −66, 82}    -   {45, −90, 42, 50, −90, 38, 53, −89, 34, 56, −88, 30, 60, −88,        26, 63, −86, 21, 66, −85, 17, 68, −84, 13, 72, −82, 9, 74, −80,        4, 77, −78}    -   {42, −89, 60, 21, −84, 74, 0, −74, 84, −21, −60, 89, −42, −42,        89, −60, −21, 84, −74, 0, 74, −84, 21, 60, −89, 42, 42, −89, 60,        21, −84, 74}    -   {38, −86, 74, −9, −63, 90, −53, −21, 80, −82, 26, 50, −89, 66,        4, −72, 88, −42, −34, 85, −77, 13, 60, −90, 56, 17, −78, 84,        −30, −45, 88, −68}    -   {34, −82, 84, −38, −30, 80, −85, 42, 26, −78, 86, −45, −21, 77,        −88, 50, 17, −74, 88, −53, −13, 72, −89, 56, 9, −68, 90, −60,        −4, 66, −90, 63}    -   {30, −77, 89, −63, 9, 50, −85, 84, −45, −13, 66, −90, 74, −26,        −34, 78, −88, 60, −4, −53, 86, −82, 42, 17, −68, 90, −72, 21,        38, −80, 88, −56}    -   {26, −68, 89, −80, 45, 4, −53, 84, −88, 63, −17, −34, 74, −90,        77, −38, −13, 60, −86, 85, −56, 9, 42, −78, 90, −72, 30, 21,        −66, 88, −82, 50}    -   {21, −60, 84, −89, 74, −42, 0, 42, −74, 89, −84, 60, −21, −21,        60, −84, 89, −74, 42, 0, −42, 74, −89, 84, −60, 21, 21, −60, 84,        −89, 74, −42}    -   {17, −50, 74, −88, 88, −77, 53, −21, −13, 45, −72, 86, −89, 78,        −56, 26, 9, −42, 68, −85, 90, −80, 60, −30, −4, 38, −66, 84,        −90, 82, −63, 34}    -   {13, −38, 60, −77, 86, −90, 85, −74, 56, −34, 9, 17, −42, 63,        −78, 88, −90, 84, −72, 53, −30, 4, 21, −45, 66, −80, 88, −89,        82, −68, 50, −26}    -   {9, −26, 42, −56, 68, −78, 85, −89, 90, −86, 80, −72, 60, −45,        30, −13, −4, 21, −38, 53, −66, 77, −84, 88, −90, 88, −82, 74,        −63, 50, −34, 17}    -   {4, −13, 21, −30, 38, −45, 53, −60, 66, −72, 77, −80, 84, −86,        88, −90, 90, −89, 88, −85, 82, −78, 74, −68, 63, −56, 50, −42,        34, −26, 17, −9}

64-Point DCT-8:

-   -   {90, 90, 90, 90, 90, 89, 90, 89, 89, 88, 87, 87, 86, 85, 84, 85,        83, 82, 80, 81, 79, 78, 77, 76, 73, 74, 72, 72, 69, 68, 66, 65,        64, 62, 60, 59, 58, 55, 53, 52, 49, 48, 46, 44, 42, 41, 38, 37,        34, 32, 30, 29, 26, 23, 22, 20, 17, 15, 13, 11, 8, 7, 5, 2}    -   {90, 90, 89, 87, 85, 83, 81, 77, 74, 69, 65, 60, 55, 49, 44, 38,        32, 26, 20, 13, 7, 0, −7, −13, −20, −26, −32, −38, −44, −49,        −55, −60, −65, −69, −74, −77, −81, −83, −85, −87, −89, −90, −90,        −90, −90, −89, −87, −85, −83, −81, −77, −74, −69, −65, −60, −55,        −49, −44, −38, −32, −26, −20, −13, −7}    -   {90, 89, 86, 82, 77, 72, 64, 55, 46, 37, 26, 15, 5, −7, −17,        −29, −38, −48, −58, −65, −72, −78, −83, −87, −90, −90, −90, −89,        −85, −80, −76, −69, −62, −53, −44, −34, −23, −13, −2, 8, 20, 30,        41, 49, 59, 66, 74, 79, 85, 87, 89, 90, 90, 88, 84, 81, 73, 68,        60, 52, 42, 32, 22, 11}    -   {90, 87, 82, 73, 65, 53, 41, 26, 11, −5, −20, −34, −48, −60,        −72, −79, −85, −90, −90, −89, −84, −78, −69, −59, −46, −32, −17,        −2, 13, 29, 42, 55, 66, 76, 83, 88, 90, 90, 87, 80, 74, 64, 52,        38, 23, 8, −7, −22, −37, −49, −62, −72, −81, −86, −89, −90, −89,        −85, −77, −68, −58, −44, −30, −15}    -   {90, 85, 77, 65, 49, 32, 13, −7, −26, −44, −60, −74, −83, −89,        −90, −87, −81, −69, −55, −38, −20, 0, 20, 38, 55, 69, 81, 87,        90, 89, 83, 74, 60, 44, 26, 7, −13, −32, −49, −65, −77, −85,        −90, −90, −85, −77, −65, −49, −32, −13, 7, 26, 44, 60, 74, 83,        89, 90, 87, 81, 69, 55, 38, 20}    -   {89, 83, 72, 53, 32, 8, −15, −38, −59, −73, −85, −90, −89, −81,        −66, −48, −26, −2, 22, 44, 64, 78, 87, 90, 87, 77, 62, 42, 20,        −5, −29, −49, −68, −80, −89, −90, −84, −74, −58, −37, −13, 11,        34, 55, 72, 85, 90, 90, 82, 69, 52, 30, 7, −17, −41, −60, −76,        −86, −90, −88, −79, −65, −46, −23}    -   {90, 81, 64, 41, 13, −15, −42, −65, −80, −89, −89, −79, −62,        −38, −11, 17, 44, 66, 82, 90, 89, 78, 60, 37, 8, −20, −46, −68,        −83, −90, −88, −77, −59, −34, −7, 22, 48, 69, 85, 90, 87, 76,        58, 32, 5, −23, −49, −72, −84, −90, −87, −73, −55, −30, −2, 26,        52, 72, 85, 90, 86, 74, 53, 29}    -   {89, 77, 55, 26, −7, −38, −65, −83, −90, −85, −69, −44, −13, 20,        49, 74, 87, 90, 81, 60, 32, 0, −32, −60, −81, −90, −87, −74,        −49, −20, 13, 44, 69, 85, 90, 83, 65, 38, 7, −26, −55, −77, −89,        −89, −77, −55, −26, 7, 38, 65, 83, 90, 85, 69, 44, 13, −20, −49,        −74, −87, −90, −81, −60, −32}    -   {89, 74, 46, 11, −26, −59, −80, −90, −85, −64, −32, 5, 41, 69,        87, 89, 77, 52, 17, −20, −53, −78, −90, −86, −68, −38, −2, 34,        65, 84, 90, 81, 58, 23, −13, −48, −73, −89, −88, −72, −44, −8,        29, 60, 82, 90, 83, 62, 30, −7, −42, −72, −87, −90, −76, −49,        −15, 22, 55, 79, 90, 85, 66, 37}    -   {88, 69, 37, −5, −44, −73, −89, −85, −64, −29, 13, 52, 79, 90,        82, 58, 20, −22, −59, −83, −90, −78, −49, −11, 30, 65, 86, 90,        74, 42, 2, −38, −72, −89, −87, −68, −34, 7, 46, 76, 90, 84, 62,        26, −15, −53, −81, −90, −80, −55, −17, 23, 60, 85, 90, 77, 48,        8, −32, −66, −87, −89, −72, −41}    -   {87, 65, 26, −20, −60, −85, −89, −69, −32, 13, 55, 83, 90, 74,        38, −7, −49, −81, −90, −77, −44, 0, 44, 77, 90, 81, 49, 7, −38,        −74, −90, −83, −55, −13, 32, 69, 89, 85, 60, 20, −26, −65, −87,        −87, −65, −26, 20, 60, 85, 89, 69, 32, −13, −55, −83, −90, −74,        −38, 7, 49, 81, 90, 77, 44}    -   {87, 60, 15, −34, −74, −90, −79, −44, 5, 52, 83, 90, 68, 26,        −23, −66, −89, −85, −53, −7, 42, 78, 90, 73, 37, −13, −59, −86,        −87, −62, −17, 32, 72, 90, 81, 46, −2, −49, −82, −89, −69, −29,        22, 65, 89, 84, 55, 8, −41, −77, −90, −76, −38, 11, 58, 85, 88,        64, 20, −30, −72, −90, −80, −48}    -   {86, 55, 5, −48, −83, −89, −62, −13, 41, 79, 90, 68, 22, −32,        −73, −90, −74, −30, 23, 69, 90, 78, 38, −15, −64, −89, −82, −46,        7, 58, 87, 85, 53, 2, −49, −85, −88, −60, −11, 42, 81, 89, 66,        20, −34, −76, −90, −72, −29, 26, 72, 90, 77, 37, −17, −65, −90,        −80, −44, 8, 59, 87, 84, 52}    -   {85, 49, −7, −60, −89, −81, −38, 20, 69, 90, 74, 26, −32, −77,        −90, −65, −13, 44, 83, 87, 55, 0, −55, −87, −83, −44, 13, 65,        90, 77, 32, −26, −74, −90, −69, −20, 38, 81, 89, 60, 7, −49,        −85, −85, −49, 7, 60, 89, 81, 38, −20, −69, −90, −74, −26, 32,        77, 90, 65, 13, −44, −83, −87, −55}    -   {84, 44, −17, −72, −90, −66, −11, 49, 87, 82, 38, −23, −73, −90,        −62, −5, 55, 89, 79, 32, −30, −78, −89, −58, 2, 60, 89, 76, 26,        −37, −80, −87, −52, 8, 65, 90, 72, 20, −42, −85, −85, −46, 15,        69, 90, 68, 13, −48, −86, −83, −41, 22, 74, 90, 64, 7, −53, −88,        −81, −34, 29, 77, 90, 59}    -   {85, 38, −29, −79, −87, −48, 17, 74, 89, 58, −7, −66, −90, −65,        −5, 59, 90, 72, 15, −49, −88, −78, −26, 41, 84, 83, 37, −30,        −81, −87, −46, 20, 73, 90, 55, −8, −68, −90, −64, −2, 60, 90,        72, 13, −52, −89, −77, −23, 42, 85, 82, 34, −32, −80, −86, −44,        22, 76, 89, 53, −11, −69, −90, −62}    -   {83, 32, −38, −85, −81, −26, 44, 87, 77, 20, −49, −89, −74, −13,        55, 90, 69, 7, −60, −90, −65, 0, 65, 90, 60, −7, −69, −90, −55,        13, 74, 89, 49, −20, −77, −87, −44, 26, 81, 85, 38, −32, −83,        −83, −32, 38, 85, 81, 26, −44, −87, −77, −20, 49, 89, 74, 13,        −55, −90, −69, −7, 60, 90, 65}    -   {82, 26, −48, −90, −69, −2, 66, 90, 52, −22, −81, −85, −30, 44,        89, 72, 7, −64, −90, −55, 17, 78, 85, 34, −41, −87, −73, −11,        60, 90, 59, −13, −76, −87, −38, 37, 86, 77, 15, −58, −90, −62,        8, 74, 88, 42, −32, −84, −79, −20, 53, 90, 65, −5, −72, −89,        −46, 29, 83, 80, 23, −49, −89, −68}    -   {80, 20, −58, −90, −55, 22, 82, 81, 17, −59, −90, −53, 23, 83,        79, 15, −60, −90, −52, 26, 85, 78, 13, −62, −90, −49, 29, 84,        77, 11, −64, −90, −48, 30, 85, 76, 8, −65, −89, −46, 32, 86, 73,        7, −66, −90, −44, 34, 87, 74, 5, −68, −89, −42, 37, 87, 72, 2,        −69, −89, −41, 38, 88, 72}    -   {81, 13, −65, −89, −38, 44, 90, 60, −20, −83, −77, −7, 69, 87,        32, −49, −90, −55, 26, 85, 74, 0, −74, −85, −26, 55, 90, 49,        −32, −87, −69, 7, 77, 83, 20, −60, −90, −44, 38, 89, 65, −13,        −81, −81, −13, 65, 89, 38, −44, −90, −60, 20, 83, 77, 7, −69,        −87, −32, 49, 90, 55, −26, −85, −74}    -   {79, 7, −72, −84, −20, 64, 89, 32, −53, −90, −44, 42, 90, 55,        −30, −88, −65, 17, 85, 74, −5, −78, −81, −8, 72, 85, 22, −62,        −89, −34, 52, 90, 46, −41, −90, −58, 29, 87, 66, −15, −83, −73,        2, 77, 80, 11, −69, −86, −23, 60, 90, 37, −49, −90, −48, 38, 89,        59, −26, −87, −68, 13, 82, 76}    -   {78, 0, −78, −78, 0, 78, 78, 0, −78, −78, 0, 78, 78, 0, −78,        −78, 0, 78, 78, 0, −78, −78, 0, 78, 78, 0, −78, −78, 0, 78, 78,        0, −78, −78, 0, 78, 78, 0, −78, −78, 0, 78, 78, 0, −78, −78, 0,        78, 78, 0, −78, −78, 0, 78, 78, 0, −78, −78, 0, 78, 78, 0, −78,        −78}    -   {77, −7, −83, −69, 20, 87, 60, −32, −90, −49, 44, 90, 38, −55,        −89, −26, 65, 85, 13, −74, −81, 0, 81, 74, −13, −85, −65, 26,        89, 55, −38, −90, −44, 49, 90, 32, −60, −87, −20, 69, 83, 7,        −77, −77, 7, 83, 69, −20, −87, −60, 32, 90, 49, −44, −90, −38,        55, 89, 26, −65, −85, −13, 74, 81}    -   {76, −13, −87, −59, 38, 90, 37, −60, −86, −11, 77, 73, −15, −87,        −58, 41, 90, 34, −62, −85, −8, 78, 74, −17, −88, −55, 42, 90,        32, −64, −84, −7, 79, 72, −20, −89, −53, 44, 90, 30, −65, −85,        −5, 81, 72, −22, −89, −52, 46, 90, 29, −66, −83, −2, 80, 69,        −23, −90, −49, 48, 89, 26, −68, −82}    -   {73, −20, −90, −46, 55, 87, 8, −81, −68, 30, 90, 37, −64, −83,        2, 84, 60, −41, −90, −26, 72, 78, −13, −88, −52, 49, 89, 15,        −77, −72, 23, 90, 42, −59, −85, −5, 82, 65, −34, −90, −32, 66,        80, −7, −86, −58, 44, 89, 22, −74, −76, 17, 89, 48, −53, −87,        −11, 79, 69, −29, −90, −38, 62, 85}    -   {74, −26, −90, −32, 69, 77, −20, −90, −38, 65, 81, −13, −89,        −44, 60, 83, −7, −87, −49, 55, 85, 0, −85, −55, 49, 87, 7, −83,        −60, 44, 89, 13, −81, −65, 38, 90, 20, −77, −69, 32, 90, 26,        −74, −74, 26, 90, 32, −69, −77, 20, 90, 38, −65, −81, 13, 89,        44, −60, −83, 7, 87, 49, −55, −85}    -   {72, −32, −90, −17, 81, 62, −46, −87, −2, 86, 49, −59, −82, 13,        89, 37, −69, −73, 29, 90, 22, −78, −65, 42, 89, 7, −84, −53, 55,        85, −8, −89, −41, 66, 77, −23, −90, −26, 76, 68, −38, −90, −11,        83, 58, −52, −85, 5, 88, 44, −64, −79, 20, 90, 30, −74, −72, 34,        90, 15, −80, −60, 48, 87}    -   {72, −38, −89, −2, 87, 42, −68, −74, 34, 90, 7, −86, −46, 65,        76, −30, −90, −11, 84, 49, −62, −78, 26, 90, 15, −83, −53, 59,        81, −22, −90, −20, 80, 58, −55, −82, 17, 90, 23, −79, −60, 52,        85, −13, −90, −29, 77, 64, −48, −85, 8, 89, 32, −73, −66, 44,        87, −5, −89, −37, 72, 69, −41, −88}    -   {69, −44, −85, 13, 90, 20, −83, −49, 65, 74, −38, −87, 7, 90,        26, −81, −55, 60, 77, −32, −89, 0, 89, 32, −77, −60, 55, 81,        −26, −90, −7, 87, 38, −74, −65, 49, 83, −20, −90, −13, 85, 44,        −69, −69, 44, 85, −13, −90, −20, 83, 49, −65, −74, 38, 87, −7,        −90, −26, 81, 55, −60, −77, 32, 89}    -   {68, −49, −80, 29, 89, −5, −90, −20, 84, 42, −74, −62, 58, 77,        −37, −87, 13, 90, 11, −87, −34, 78, 55, −64, −72, 44, 85, −22,        −90, −2, 90, 26, −82, −48, 69, 66, −52, −81, 30, 89, −7, −90,        −17, 85, 41, −73, −60, 59, 76, −38, −86, 15, 90, 8, −88, −32,        79, 53, −65, −72, 46, 83, −23, −89}    -   {66, −55, −76, 42, 83, −29, −88, 13, 90, 2, −90, −17, 87, 32,        −80, −46, 74, 59, −64, −69, 52, 78, −38, −84, 23, 89, −8, −90,        −7, 90, 22, −85, −37, 79, 49, −72, −62, 60, 72, −48, −81, 34,        86, −20, −89, 5, 90, 11, −89, −26, 85, 41, −77, −53, 68, 65,        −58, −73, 44, 82, −30, −87, 15, 90}    -   {65, −60, −69, 55, 74, −49, −77, 44, 81, −38, −83, 32, 85, −26,        −87, 20, 89, −13, −90, 7, 90, 0, −90, −7, 90, 13, −89, −20, 87,        26, −85, −32, 83, 38, −81, −44, 77, 49, −74, −55, 69, 60, −65,        −65, 60, 69, −55, −74, 49, 77, −44, −81, 38, 83, −32, −85, 26,        87, −20, −89, 13, 90, −7, −90}    -   {64, −65, −62, 66, 60, −68, −59, 69, 58, −72, −55, 72, 53, −74,        −52, 73, 49, −76, −48, 77, 46, −78, −44, 79, 42, −81, −41, 80,        38, −82, −37, 83, 34, −85, −32, 84, 30, −85, −29, 86, 26, −87,        −23, 87, 22, −88, −20, 89, 17, −89, −15, 90, 13, −89, −11, 90,        8, −90, −7, 90, 5, −90, −2, 90}    -   {62, −69, −53, 76, 44, −80, −34, 85, 23, −89, −13, 90, 2, −90,        8, 90, −20, −87, 30, 83, −41, −78, 49, 72, −59, −65, 66, 58,        −74, −48, 79, 38, −85, −29, 87, 17, −89, −7, 90, −5, −90, 15,        88, −26, −84, 37, 81, −46, −73, 55, 68, −64, −60, 72, 52, −77,        −42, 82, 32, −86, −22, 89, 11, −90}    -   {60, −74, −44, 83, 26, −89, −7, 90, −13, −87, 32, 81, −49, −69,        65, 55, −77, −38, 85, 20, −90, 0, 90, −20, −85, 38, 77, −55,        −65, 69, 49, −81, −32, 87, 13, −90, 7, 89, −26, −83, 44, 74,        −60, −60, 74, 44, −83, −26, 89, 7, −90, 13, 87, −32, −81, 49,        69, −65, −55, 77, 38, −85, −20, 90}    -   {59, −77, −34, 88, 7, −90, 22, 83, −48, −68, 69, 46, −85, −20,        90, −8, −87, 37, 76, −60, −58, 78, 32, −89, −5, 90, −23, −82,        49, 66, −72, −44, 84, 17, −90, 11, 87, −38, −73, 62, 55, −79,        −30, 89, 2, −89, 26, 80, −52, −65, 72, 42, −85, −15, 90, −13,        −86, 41, 74, −64, −53, 81, 29, −90}    -   {58, −81, −23, 90, −13, −84, 48, 65, −73, −34, 89, −2, −88, 38,        72, −68, −44, 86, 8, −90, 29, 78, −60, −53, 82, 20, −90, 17, 83,        −52, −62, 77, 30, −89, 7, 87, −42, −69, 72, 41, −87, −5, 90,        −32, −76, 64, 49, −85, −15, 90, −22, −80, 55, 59, −79, −26, 90,        −11, −85, 46, 66, −74, −37, 89}    -   {55, −83, −13, 90, −32, −74, 69, 38, −89, 7, 85, −49, −60, 81,        20, −90, 26, 77, −65, −44, 87, 0, −87, 44, 65, −77, −26, 90,        −20, −81, 60, 49, −85, −7, 89, −38, −69, 74, 32, −90, 13, 83,        −55, −55, 83, 13, −90, 32, 74, −69, −38, 89, −7, −85, 49, 60,        −81, −20, 90, −26, −77, 65, 44, −87}    -   {53, −85, −2, 87, −49, −58, 85, 7, −88, 46, 60, −82, −11, 89,        −42, −64, 81, 15, −89, 38, 66, −78, −20, 90, −34, −69, 76, 23,        −90, 30, 72, −74, −29, 90, −26, −73, 72, 32, −90, 22, 77, −68,        −37, 90, −17, −79, 65, 41, −90, 13, 80, −62, −44, 89, −8, −83,        59, 48, −87, 5, 84, −55, −52, 86}    -   {52, −87, 8, 80, −65, −37, 90, −26, −72, 76, 20, −89, 42, 60,        −85, −2, 85, −58, −46, 89, −15, −78, 69, 30, −90, 32, 68, −79,        −13, 89, −48, −55, 86, −5, −83, 62, 41, −90, 22, 73, −74, −23,        90, −38, −64, 82, 7, −87, 53, 49, −88, 11, 81, −66, −34, 90,        −29, −72, 77, 17, −90, 44, 59, −84}    -   {49, −89, 20, 74, −77, −13, 87, −55, −44, 90, −26, −69, 81, 7,        −85, 60, 38, −90, 32, 65, −83, 0, 83, −65, −32, 90, −38, −60,        85, −7, −81, 69, 26, −90, 44, 55, −87, 13, 77, −74, −20, 89,        −49, −49, 89, −20, −74, 77, 13, −87, 55, 44, −90, 26, 69, −81,        −7, 85, −60, −38, 90, −32, −65, 83}    -   {48, −90, 30, 64, −85, 11, 76, −77, −8, 84, −65, −29, 89, −49,        −46, 90, −32, −62, 86, −13, −73, 78, 7, −85, 66, 26, −90, 52,        44, −90, 34, 60, −87, 15, 74, −79, −5, 83, −68, −23, 89, −53,        −42, 90, −37, −59, 87, −17, −72, 81, 2, −82, 69, 22, −89, 55,        41, −90, 38, 58, −88, 20, 72, −80}    -   {46, −90, 41, 52, −90, 34, 58, −89, 29, 62, −87, 22, 66, −85,        15, 72, −83, 8, 73, −81, 2, 78, −77, −5, 80, −74, −11, 85, −69,        −17, 86, −65, −23, 88, −60, −30, 90, −55, −37, 90, −49, −42, 90,        −44, −48, 90, −38, −53, 89, −32, −59, 89, −26, −64, 87, −20,        −68, 84, −13, −72, 82, −7, −76, 79}    -   {44, −90, 49, 38, −90, 55, 32, −89, 60, 26, −87, 65, 20, −85,        69, 13, −83, 74, 7, −81, 77, 0, −77, 81, −7, −74, 83, −13, −69,        85, −20, −65, 87, −26, −60, 89, −32, −55, 90, −38, −49, 90, −44,        −44, 90, −49, −38, 90, −55, −32, 89, −60, −26, 87, −65, −20, 85,        −69, −13, 83, −74, −7, 81, −77}    -   {42, −90, 59, 23, −85, 72, 5, −77, 82, −15, −65, 89, −34, −49,        90, −52, −32, 88, −66, −13, 80, −78, 7, 72, −86, 26, 58, −90,        44, 41, −89, 60, 22, −84, 74, 2, −76, 83, −17, −64, 89, −37,        −48, 90, −53, −30, 87, −68, −11, 81, −79, 8, 69, −87, 29, 55,        −90, 46, 38, −90, 62, 20, −85, 73}    -   {41, −89, 66, 8, −77, 85, −23, −55, 90, −53, −26, 84, −76, 7,        68, −89, 38, 42, −90, 65, 11, −78, 83, −22, −58, 90, −52, −29,        85, −73, 5, 69, −88, 37, 44, −89, 64, 13, −79, 82, −20, −59, 90,        −49, −30, 86, −74, 2, 72, −87, 34, 46, −90, 62, 15, −81, 80,        −17, −60, 90, −48, −32, 87, −72}    -   {38, −87, 74, −7, −65, 90, −49, −26, 83, −81, 20, 55, −90, 60,        13, −77, 85, −32, −44, 89, −69, 0, 69, −89, 44, 32, −85, 77,        −13, −60, 90, −55, −20, 81, −83, 26, 49, −90, 65, 7, −74, 87,        −38, −38, 87, −74, 7, 65, −90, 49, 26, −83, 81, −20, −55, 90,        −60, −13, 77, −85, 32, 44, −89, 69}    -   {37, −85, 79, −22, −49, 90, −72, 7, 62, −90, 60, 8, −72, 89,        −48, −23, 81, −84, 34, 38, −86, 78, −20, −52, 89, −69, 5, 64,        −90, 59, 11, −74, 89, −46, −26, 80, −85, 32, 41, −87, 77, −17,        −53, 90, −68, 2, 65, −90, 58, 13, −73, 88, −44, −29, 82, −83,        30, 42, −87, 76, −15, −55, 90, −66}    -   {34, −83, 85, −37, −32, 82, −84, 38, 30, −80, 85, −41, −29, 81,        −86, 42, 26, −79, 87, −44, −23, 78, −87, 46, 22, −77, 88, −48,        −20, 76, −89, 49, 17, −73, 89, −52, −15, 74, −90, 53, 13, −72,        89, −55, −11, 72, −90, 58, 8, −69, 90, −59, −7, 68, −90, 60, 5,        −66, 90, −62, −2, 65, −90, 64}    -   {32, −81, 87, −49, −13, 69, −90, 65, −7, −55, 89, −77, 26, 38,        −83, 85, −44, −20, 74, −90, 60, 0, −60, 90, −74, 20, 44, −85,        83, −38, −26, 77, −89, 55, 7, −65, 90, −69, 13, 49, −87, 81,        −32, −32, 81, −87, 49, 13, −69, 90, −65, 7, 55, −89, 77, −26,        −38, 83, −85, 44, 20, −74, 90, −60}    -   {30, −77, 89, −62, 7, 52, −87, 83, −42, −17, 69, −90, 72, −20,        −41, 82, −87, 53, 5, −60, 90, −78, 32, 29, −76, 90, −64, 8, 49,        −86, 85, −44, −15, 68, −90, 72, −22, −38, 80, −88, 55, 2, −59,        89, −79, 34, 26, −73, 90, −65, 11, 48, −85, 84, −46, −13, 66,        −90, 74, −23, −37, 81, −89, 58}    -   {29, −74, 90, −72, 26, 30, −73, 90, −72, 23, 32, −76, 90, −69,        22, 34, −77, 90, −68, 20, 37, −78, 90, −66, 17, 38, −79, 89,        −65, 15, 41, −81, 90, −64, 13, 42, −80, 89, −62, 11, 44, −82,        89, −60, 8, 46, −83, 88, −59, 7, 48, −85, 87, −58, 5, 49, −84,        87, −55, 2, 52, −85, 86, −53}    -   {26, −69, 90, −81, 44, 7, −55, 85, −87, 60, −13, −38, 77, −90,        74, −32, −20, 65, −89, 83, −49, 0, 49, −83, 89, −65, 20, 32,        −74, 90, −77, 38, 13, −60, 87, −85, 55, −7, −44, 81, −90, 69,        −26, −26, 69, −90, 81, −44, −7, 55, −85, 87, −60, 13, 38, −77,        90, −74, 32, 20, −65, 89, −83, 49}    -   {23, −65, 88, −86, 60, −17, −30, 69, −90, 85, −55, 11, 37, −74,        90, −80, 49, −5, −42, 77, −90, 78, −44, −2, 48, −81, 90, −73,        38, 8, −53, 83, −89, 72, −32, −15, 59, −85, 89, −66, 26, 22,        −64, 87, −87, 62, −20, −29, 68, −89, 84, −58, 13, 34, −72, 90,        −82, 52, −7, −41, 76, −90, 79, −46}    -   {22, −60, 84, −89, 74, −41, −2, 44, −76, 90, −83, 58, −17, −26,        64, −86, 89, −72, 37, 7, −48, 78, −90, 80, −53, 13, 30, −66, 87,        −88, 68, −32, −11, 52, −81, 90, −79, 49, −8, −34, 69, −89, 87,        −65, 29, 15, −55, 82, −90, 77, −46, 5, 38, −72, 90, −85, 62,        −23, −20, 59, −85, 90, −73, 42}    -   {20, −55, 81, −90, 83, −60, 26, 13, −49, 77, −90, 85, −65, 32,        7, −44, 74, −89, 87, −69, 38, 0, −38, 69, −87, 89, −74, 44, −7,        −32, 65, −85, 90, −77, 49, −13, −26, 60, −83, 90, −81, 55, −20,        −20, 55, −81, 90, −83, 60, −26, −13, 49, −77, 90, −85, 65, −32,        −7, 44, −74, 89, −87, 69, −38}    -   {17, −49, 73, −89, 89, −76, 52, −20, −15, 48, −74, 88, −90, 77,        −53, 22, 13, −46, 72, −87, 89, −78, 55, −23, −11, 44, −72, 87,        −90, 79, −58, 26, 8, −42, 69, −86, 90, −81, 59, −29, −7, 41,        −68, 85, −90, 80, −60, 30, 5, −38, 66, −84, 90, −82, 62, −32,        −2, 37, −65, 85, −90, 83, −64, 34}    -   {15, −44, 68, −85, 90, −86, 72, −49, 22, 8, −38, 64, −80, 90,        −88, 76, −55, 29, 2, −32, 59, −78, 89, −90, 79, −60, 34, −5,        −26, 53, −73, 87, −90, 82, −65, 41, −11, −20, 48, −72, 85, −90,        84, −69, 46, −17, −13, 42, −66, 83, −90, 87, −74, 52, −23, −7,        37, −62, 81, −89, 89, −77, 58, −30}    -   {13, −38, 60, −77, 87, −90, 85, −74, 55, −32, 7, 20, −44, 65,        −81, 89, −90, 83, −69, 49, −26, 0, 26, −49, 69, −83, 90, −89,        81, −65, 44, −20, −7, 32, −55, 74, −85, 90, −87, 77, −60, 38,        −13, −13, 38, −60, 77, −87, 90, −85, 74, −55, 32, −7, −20, 44,        −65, 81, −89, 90, −83, 69, −49, 26}    -   {11, −32, 52, −68, 81, −88, 90, −87, 79, −66, 49, −30, 8, 13,        −34, 53, −69, 80, −89, 90, −87, 78, −65, 48, −29, 7, 15, −37,        55, −72, 82, −89, 90, −86, 77, −64, 46, −26, 5, 17, −38, 58,        −72, 83, −90, 90, −85, 76, −62, 44, −23, 2, 20, −41, 59, −74,        85, −89, 90, −84, 73, −60, 42, −22}    -   {8, −26, 42, −58, 69, −79, 86, −90, 90, −87, 81, −72, 59, −44,        29, −11, −7, 23, −41, 55, −68, 78, −85, 89, −90, 87, −80, 72,        −60, 46, −30, 13, 5, −22, 38, −53, 66, −77, 84, −90, 90, −88,        82, −74, 62, −48, 32, −15, −2, 20, −37, 52, −65, 76, −85, 89,        −90, 89, −83, 73, −64, 49, −34, 17}    -   {7, −20, 32, −44, 55, −65, 74, −81, 85, −89, 90, −90, 87, −83,        77, −69, 60, −49, 38, −26, 13, 0, −13, 26, −38, 49, −60, 69,        −77, 83, −87, 90, −90, 89, −85, 81, −74, 65, −55, 44, −32, 20,        −7, −7, 20, −32, 44, −55, 65, −74, 81, −85, 89, −90, 90, −87,        83, −77, 69, −60, 49, −38, 26, −13}    -   {5, −13, 22, −30, 38, −46, 53, −60, 66, −72, 77, −80, 84, −87,        90, −90, 90, −89, 88, −85, 82, −78, 74, −68, 62, −55, 48, −41,        32, −23, 15, −7, −2, 11, −20, 29, −37, 44, −52, 59, −65, 72,        −76, 81, −85, 87, −89, 90, −90, 90, −89, 86, −83, 79, −73, 69,        −64, 58, −49, 42, −34, 26, −17, 8}    -   {2, −7, 11, −15, 20, −23, 29, −32, 37, −41, 44, −48, 52, −55,        59, −62, 65, −68, 72, −74, 76, −78, 81, −82, 85, −85, 87, −88,        89, −89, 90, −90, 90, −90, 90, −90, 89, −87, 86, −84, 83, −80,        79, −77, 73, −72, 69, −66, 64, −60, 58, −53, 49, −46, 42, −38,        34, −30, 26, −22, 17, −13, 8, −5}

After minimizing the cost value by trying all possible combination ofoffset values on a selected set of M unique numbers, T2 may be output asthe B-bit N-point primary transform core.

When deriving the 2N-point DST-1 transform cores, it is restricted thatthe set of M unique numbers which construct the 2N-point DST-1 corematrix is the same set of M unique numbers which construct the N-pointDST-7 and/or N-point DCT-8 core matrix. In one embodiment, M equals toN. In one embodiment, one of the 8-bit DST-1 transform cores shown belowmay be used. For example, in 8-point DST-1, there are 4 unique numbersincluding 28, 55, 74, 84, and this is the same set of 4 unique numbersused to construct 4-point DST-7 and 4-point DCT-8 cores using theembodiment where M equals N discussed above.

The 8-bit DST-1 transform cores may include:

4-Point DST-1:

-   -   {48, 77, 77, 48}    -   {77, 48, −48, −77}    -   {77, −48, −48, 77}    -   {48, −77, 77, −48}

8-Point DST-1:

-   -   {29, 55, 74, 84, 84, 74, 55, 29}    -   {55, 84, 74, 29, −29, −74, −84, −55}    -   {74, 74, 0, −74, −74, 0, 74, 74}    -   {84, 29, −74, −55, 55, 74, −29, −84}    -   {84, −29, −74, 55, 55, −74, −29, 84}    -   {74, −74, 0, 74, −74, 0, 74, −74}    -   {55, −84, 74, −29, −29, 74, −84, 55}    -   {29, −55, 74, −84, 84, −74, 55, −29}

16-Point DST-1:

-   -   {17, 32, 46, 60, 71, 78, 85, 86, 86, 85, 78, 71, 60, 46, 32, 17}    -   {32, 60, 78, 86, 85, 71, 46, 17, −17, −46, −71, −85, −86, −78,        −60, −32}    -   {46, 78, 86, 71, 32, −17, −60, −85, −85, −60, −17, 32, 71, 86,        78, 46}    -   {60, 86, 71, 17, −46, −85, −78, −32, 32, 78, 85, 46, −17, −71,        −86, −60}    -   {71, 85, 32, −46, −86, −60, 17, 78, 78, 17, −60, −86, −46, 32,        85, 71}    -   {78, 71, −17, −85, −60, 32, 86, 46, −46, −86, −32, 60, 85, 17,        −71, −78}    -   {85, 46, −60, −78, 17, 86, 32, −71, −71, 32, 86, 17, −78, −60,        46, 85}    -   {86, 17, −85, −32, 78, 46, −71, −60, 60, 71, −46, −78, 32, 85,        −17, −86}    -   {86, −17, −85, 32, 78, −46, −71, 60, 60, −71, −46, 78, 32, −85,        −17, 86}    -   {85, −46, −60, 78, 17, −86, 32, 71, −71, −32, 86, −17, −78, 60,        46, −85}    -   {78, −71, −17, 85, −60, −32, 86, −46, −46, 86, −32, −60, 85,        −17, −71, 78}    -   {71, −85, 32, 46, −86, 60, 17, −78, 78, −17, −60, 86, −46, −32,        85, −71}    -   {60, −86, 71, −17, −46, 85, −78, 32, 32, −78, 85, −46, −17, 71,        −86, 60}    -   {46, −78, 86, −71, 32, 17, −60, 85, −85, 60, −17, −32, 71, −86,        78, −46}    -   {32, −60, 78, −86, 85, −71, 46, −17, −17, 46, −71, 85, −86, 78,        −60, 32}    -   {17, −32, 46, −60, 71, −78, 85, −86, 86, −85, 78, −71, 60, −46,        32, −17}

32-Point DST-1:

-   -   {9, 17, 25, 33, 41, 49, 56, 62, 66, 72, 77, 81, 83, 87, 89, 90,        90, 89, 87, 83, 81, 77, 72, 66, 62, 56, 49, 41, 33, 25, 17, 9}    -   {17, 33, 49, 62, 72, 81, 87, 90, 89, 83, 77, 66, 56, 41, 25, 9,        −9, −25, −41, −56, −66, −77, −83, −89, −90, −87, −81, −72, −62,        −49, −33, −17}    -   {25, 49, 66, 81, 89, 89, 81, 66, 49, 25, 0, −25, −49, −66, −81,        −89, −89, −81, −66, −49, −25, 0, 25, 49, 66, 81, 89, 89, 81, 66,        49, 25}    -   {33, 62, 81, 90, 83, 66, 41, 9, −25, −56, −77, −89, −87, −72,        −49, −17, 17, 49, 72, 87, 89, 77, 56, 25, −9, −41, −66, −83,        −90, −81, −62, −33}    -   {41, 72, 89, 83, 62, 25, −17, −56, −81, −90, −77, −49, −9, 33,        66, 87, 87, 66, 33, −9, −49, −77, −90, −81, −56, −17, 25, 62,        83, 89, 72, 41}    -   {49, 81, 89, 66, 25, −25, −66, −89, −81, −49, 0, 49, 81, 89, 66,        25, −25, −66, −89, −81, −49, 0, 49, 81, 89, 66, 25, −25, −66,        −89, −81, −49}    -   {56, 87, 81, 41, −17, −66, −90, −72, −25, 33, 77, 89, 62, 9,        −49, −83, −83, −49, 9, 62, 89, 77, 33, −25, −72, −90, −66, −17,        41, 81, 87, 56}    -   {62, 90, 66, 9, −56, −89, −72, −17, 49, 87, 77, 25, −41, −83,        −81, −33, 33, 81, 83, 41, −25, −77, −87, −49, 17, 72, 89, 56,        −9, −66, −90, −62}    -   {66, 89, 49, −25, −81, −81, −25, 49, 89, 66, 0, −66, −89, −49,        25, 81, 81, 25, −49, −89, −66, 0, 66, 89, 49, −25, −81, −81,        −25, 49, 89, 66}    -   {72, 83, 25, −56, −90, −49, 33, 87, 66, −9, −77, −81, −17, 62,        89, 41, −41, −89, −62, 17, 81, 77, 9, −66, −87, −33, 49, 90, 56,        −25, −83, −72}    -   {77, 77, 0, −77, −77, 0, 77, 77, 0, −77, −77, 0, 77, 77, 0, −77,        −77, 0, 77, 77, 0, −77, −77, 0, 77, 77, 0, −77, −77, 0, 77, 77}    -   {81, 66, −25, −89, −49, 49, 89, 25, −66, −81, 0, 81, 66, −25,        −89, −49, 49, 89, 25, −66, −81, 0, 81, 66, −25, −89, −49, 49,        89, 25, −66, −81}    -   {83, 56, −49, −87, −9, 81, 62, −41, −89, −17, 77, 66, −33, −90,        −25, 72, 72, −25, −90, −33, 66, 77, −17, −89, −41, 62, 81, −9,        −87, −49, 56, 83}    -   {87, 41, −66, −72, 33, 89, 9, −83, −49, 62, 77, −25, −90, −17,        81, 56, −56, −81, 17, 90, 25, −77, −62, 49, 83, −9, −89, −33,        72, 66, −41, −87}    -   {89, 25, −81, −49, 66, 66, −49, −81, 25, 89, 0, −89, −25, 81,        49, −66, −66, 49, 81, −25, −89, 0, 89, 25, −81, −49, 66, 66,        −49, −81, 25, 89}    -   {90, 9, −89, −17, 87, 25, −83, −33, 81, 41, −77, −49, 72, 56,        −66, −62, 62, 66, −56, −72, 49, 77, −41, −81, 33, 83, −25, −87,        17, 89, −9, −90}    -   {90, −9, −89, 17, 87, −25, −83, 33, 81, −41, −77, 49, 72, −56,        −66, 62, 62, −66, −56, 72, 49, −77, −41, 81, 33, −83, −25, 87,        17, −89, −9, 90}    -   {89, −25, −81, 49, 66, −66, −49, 81, 25, −89, 0, 89, −25, −81,        49, 66, −66, −49, 81, 25, −89, 0, 89, −25, −81, 49, 66, −66,        −49, 81, 25, −89}    -   {87, −41, −66, 72, 33, −89, 9, 83, −49, −62, 77, 25, −90, 17,        81, −56, −56, 81, 17, −90, 25, 77, −62, −49, 83, 9, −89, 33, 72,        −66, −41, 87}    -   {83, −56, −49, 87, −9, −81, 62, 41, −89, 17, 77, −66, −33, 90,        −25, −72, 72, 25, −90, 33, 66, −77, −17, 89, −41, −62, 81, 9,        −87, 49, 56, −83}    -   {81, −66, −25, 89, −49, −49, 89, −25, −66, 81, 0, −81, 66, 25,        −89, 49, 49, −89, 25, 66, −81, 0, 81, −66, −25, 89, −49, −49,        89, −25, −66, 81}    -   {77, −77, 0, 77, −77, 0, 77, −77, 0, 77, −77, 0, 77, −77, 0, 77,        −77, 0, 77, −77, 0, 77, −77, 0, 77, −77, 0, 77, −77, 0, 77, −77}    -   {72, −83, 25, 56, −90, 49, 33, −87, 66, 9, −77, 81, −17, −62,        89, −41, −41, 89, −62, −17, 81, −77, 9, 66, −87, 33, 49, −90,        56, 25, −83, 72}    -   {66, −89, 49, 25, −81, 81, −25, −49, 89, −66, 0, 66, −89, 49,        25, −81, 81, −25, −49, 89, −66, 0, 66, −89, 49, 25, −81, 81,        −25, −49, 89, −66}    -   {62, −90, 66, −9, −56, 89, −72, 17, 49, −87, 77, −25, −41, 83,        −81, 33, 33, −81, 83, −41, −25, 77, −87, 49, 17, −72, 89, −56,        −9, 66, −90, 62}    -   {56, −87, 81, −41, −17, 66, −90, 72, −25, −33, 77, −89, 62, −9,        −49, 83, −83, 49, 9, −62, 89, −77, 33, 25, −72, 90, −66, 17, 41,        −81, 87, −56}    -   {49, −81, 89, −66, 25, 25, −66, 89, −81, 49, 0, −49, 81, −89,        66, −25, −25, 66, −89, 81, −49, 0, 49, −81, 89, −66, 25, 25,        −66, 89, −81, 49}    -   {41, −72, 89, −83, 62, −25, −17, 56, −81, 90, −77, 49, −9, −33,        66, −87, 87, −66, 33, 9, −49, 77, −90, 81, −56, 17, 25, −62, 83,        −89, 72, −41}    -   {33, −62, 81, −90, 83, −66, 41, −9, −25, 56, −77, 89, −87, 72,        −49, 17, 17, −49, 72, −87, 89, −77, 56, −25, −9, 41, −66, 83,        −90, 81, −62, 33}    -   {25, −49, 66, −81, 89, −89, 81, −66, 49, −25, 0, 25, −49, 66,        −81, 89, −89, 81, −66, 49, −25, 0, 25, −49, 66, −81, 89, −89,        81, −66, 49, −25}    -   {17, −33, 49, −62, 72, −81, 87, −90, 89, −83, 77, −66, 56, −41,        25, −9, −9, 25, −41, 56, −66, 77, −83, 89, −90, 87, −81, 72,        −62, 49, −33, 17}    -   {9, −17, 25, −33, 41, −49, 56, −62, 66, −72, 77, −81, 83, −87,        89, −90, 90, −89, 87, −83, 81, −77, 72, −66, 62, −56, 49, −41,        33, −25, 17, −9}

64-Point DST-1:

-   -   {4, 9, 13, 17, 21, 26, 30, 34, 38, 42, 45, 50, 53, 56, 60, 63,        66, 68, 72, 74, 77, 78, 80, 82, 84, 85, 86, 88, 88, 89, 90, 90,        90, 90, 89, 88, 88, 86, 85, 84, 82, 80, 78, 77, 74, 72, 68, 66,        63, 60, 56, 53, 50, 45, 42, 38, 34, 30, 26, 21, 17, 13, 9, 4}    -   {9, 17, 26, 34, 42, 50, 56, 63, 68, 74, 78, 82, 85, 88, 89, 90,        90, 88, 86, 84, 80, 77, 72, 66, 60, 53, 45, 38, 30, 21, 13, 4,        −4, −13, −21, −30, −38, −45, −53, −60, −66, −72, −77, −80, −84,        −86, −88, −90, −90, −89, −88, −85, −82, −78, −74, −68, −63, −56,        −50, −42, −34, −26, −17, −9}    -   {13, 26, 38, 50, 60, 68, 77, 82, 86, 89, 90, 88, 85, 80, 74, 66,        56, 45, 34, 21, 9, −4, −17, −30, −42, −53, −63, −72, −78, −84,        −88, −90, −90, −88, −84, −78, −72, −63, −53, −42, −30, −17, −4,        9, 21, 34, 45, 56, 66, 74, 80, 85, 88, 90, 89, 86, 82, 77, 68,        60, 50, 38, 26, 13}    -   {17, 34, 50, 63, 74, 82, 88, 90, 88, 84, 77, 66, 53, 38, 21, 4,        −13, −30, −45, −60, −72, −80, −86, −90, −89, −85, −78, −68, −56,        −42, −26, −9, 9, 26, 42, 56, 68, 78, 85, 89, 90, 86, 80, 72, 60,        45, 30, 13, −4, −21, −38, −53, −66, −77, −84, −88, −90, −88,        −82, −74, −63, −50, −34, −17}    -   {21, 42, 60, 74, 84, 89, 89, 84, 74, 60, 42, 21, 0, −21, −42,        −60, −74, −84, −89, −89, −84, −74, −60, −42, −21, 0, 21, 42, 60,        74, 84, 89, 89, 84, 74, 60, 42, 21, 0, −21, −42, −60, −74, −84,        −89, −89, −84, −74, −60, −42, −21, 0, 21, 42, 60, 74, 84, 89,        89, 84, 74, 60, 42, 21}    -   {26, 50, 68, 82, 89, 88, 80, 66, 45, 21, −4, −30, −53, −72, −84,        −90, −88, −78, −63, −42, −17, 9, 34, 56, 74, 85, 90, 86, 77, 60,        38, 13, −13, −38, −60, −77, −86, −90, −85, −74, −56, −34, −9,        17, 42, 63, 78, 88, 90, 84, 72, 53, 30, 4, −21, −45, −66, −80,        −88, −89, −82, −68, −50, −26}    -   {30, 56, 77, 88, 89, 80, 63, 38, 9, −21, −50, −72, −85, −90,        −84, −68, −45, −17, 13, 42, 66, 82, 90, 86, 74, 53, 26, −4, −34,        −60, −78, −88, −88, −78, −60, −34, −4, 26, 53, 74, 86, 90, 82,        66, 42, 13, −17, −45, −68, −84, −90, −85, −72, −50, −21, 9, 38,        63, 80, 89, 88, 77, 56, 30}    -   {34, 63, 82, 90, 84, 66, 38, 4, −30, −60, −80, −90, −85, −68,        −42, −9, 26, 56, 78, 89, 86, 72, 45, 13, −21, −53, −77, −88,        −88, −74, −50, −17, 17, 50, 74, 88, 88, 77, 53, 21, −13, −45,        −72, −86, −89, −78, −56, −26, 9, 42, 68, 85, 90, 80, 60, 30, −4,        −38, −66, −84, −90, −82, −63, −34}    -   {38, 68, 86, 88, 74, 45, 9, −30, −63, −84, −90, −78, −53, −17,        21, 56, 80, 90, 82, 60, 26, −13, −50, −77, −89, −85, −66, −34,        4, 42, 72, 88, 88, 72, 42, 4, −34, −66, −85, −89, −77, −50, −13,        26, 60, 82, 90, 80, 56, 21, −17, −53, −78, −90, −84, −63, −30,        9, 45, 74, 88, 86, 68, 38}    -   {42, 74, 89, 84, 60, 21, −21, −60, −84, −89, −74, −42, 0, 42,        74, 89, 84, 60, 21, −21, −60, −84, −89, −74, −42, 0, 42, 74, 89,        84, 60, 21, −21, −60, −84, −89, −74, −42, 0, 42, 74, 89, 84, 60,        21, −21, −60, −84, −89, −74, −42, 0, 42, 74, 89, 84, 60, 21,        −21, −60, −84, −89, −74, −42}    -   {45, 78, 90, 77, 42, −4, −50, −80, −90, −74, −38, 9, 53, 82, 89,        72, 34, −13, −56, −84, −88, −68, −30, 17, 60, 85, 88, 66, 26,        −21, −63, −86, −86, −63, −21, 26, 66, 88, 85, 60, 17, −30, −68,        −88, −84, −56, −13, 34, 72, 89, 82, 53, 9, −38, −74, −90, −80,        −50, −4, 42, 77, 90, 78, 45}    -   {50, 82, 88, 66, 21, −30, −72, −90, −78, −42, 9, 56, 85, 86, 60,        13, −38, −77, −90, −74, −34, 17, 63, 88, 84, 53, 4, −45, −80,        −89, −68, −26, 26, 68, 89, 80, 45, −4, −53, −84, −88, −63, −17,        34, 74, 90, 77, 38, −13, −60, −86, −85, −56, −9, 42, 78, 90, 72,        30, −21, −66, −88, −82, −50}    -   {53, 85, 85, 53, 0, −53, −85, −85, −53, 0, 53, 85, 85, 53, 0,        −53, −85, −85, −53, 0, 53, 85, 85, 53, 0, −53, −85, −85, −53, 0,        53, 85, 85, 53, 0, −53, −85, −85, −53, 0, 53, 85, 85, 53, 0,        −53, −85, −85, −53, 0, 53, 85, 85, 53, 0, −53, −85, −85, −53, 0,        53, 85, 85, 53}    -   {56, 88, 80, 38, −21, −72, −90, −68, −17, 42, 82, 86, 53, −4,        −60, −88, −78, −34, 26, 74, 90, 66, 13, −45, −84, −85, −50, 9,        63, 89, 77, 30, −30, −77, −89, −63, −9, 50, 85, 84, 45, −13,        −66, −90, −74, −26, 34, 78, 88, 60, 4, −53, −86, −82, −42, 17,        68, 90, 72, 21, −38, −80, −88, −56}    -   {60, 89, 74, 21, −42, −84, −84, −42, 21, 74, 89, 60, 0, −60,        −89, −74, −21, 42, 84, 84, 42, −21, −74, −89, −60, 0, 60, 89,        74, 21, −42, −84, −84, −42, 21, 74, 89, 60, 0, −60, −89, −74,        −21, 42, 84, 84, 42, −21, −74, −89, −60, 0, 60, 89, 74, 21, −42,        −84, −84, −42, 21, 74, 89, 60}    -   {63, 90, 66, 4, −60, −90, −68, −9, 56, 89, 72, 13, −53, −88,        −74, −17, 50, 88, 77, 21, −45, −86, −78, −26, 42, 85, 80, 30,        −38, −84, −82, −34, 34, 82, 84, 38, −30, −80, −85, −42, 26, 78,        86, 45, −21, −77, −88, −50, 17, 74, 88, 53, −13, −72, −89, −56,        9, 68, 90, 60, −4, −66, −90, −63}    -   {66, 90, 56, −13, −74, −88, −45, 26, 80, 84, 34, −38, −85, −78,        −21, 50, 88, 72, 9, −60, −90, −63, 4, 68, 89, 53, −17, −77, −86,        −42, 30, 82, 82, 30, −42, −86, −77, −17, 53, 89, 68, 4, −63,        −90, −60, 9, 72, 88, 50, −21, −78, −85, −38, 34, 84, 80, 26,        −45, −88, −74, −13, 56, 90, 66}    -   {68, 88, 45, −30, −84, −78, −17, 56, 90, 60, −13, −77, −85, −34,        42, 88, 72, 4, −66, −89, −50, 26, 82, 80, 21, −53, −90, −63, 9,        74, 86, 38, −38, −86, −74, −9, 63, 90, 53, −21, −80, −82, −26,        50, 89, 66, −4, −72, −88, −42, 34, 85, 77, 13, −60, −90, −56,        17, 78, 84, 30, −45, −88, −68}    -   {72, 86, 34, −45, −89, −63, 13, 78, 82, 21, −56, −90, −53, 26,        84, 77, 9, −66, −88, −42, 38, 88, 68, −4, −74, −85, −30, 50, 90,        60, −17, −80, −80, −17, 60, 90, 50, −30, −85, −74, −4, 68, 88,        38, −42, −88, −66, 9, 77, 84, 26, −53, −90, −56, 21, 82, 78, 13,        −63, −89, −45, 34, 86, 72}    -   {74, 84, 21, −60, −89, −42, 42, 89, 60, −21, −84, −74, 0, 74,        84, 21, −60, −89, −42, 42, 89, 60, −21, −84, −74, 0, 74, 84, 21,        −60, −89, −42, 42, 89, 60, −21, −84, −74, 0, 74, 84, 21, −60,        −89, −42, 42, 89, 60, −21, −84, −74, 0, 74, 84, 21, −60, −89,        −42, 42, 89, 60, −21, −84, −74}    -   {77, 80, 9, −72, −84, −17, 66, 86, 26, −60, −88, −34, 53, 90,        42, −45, −90, −50, 38, 89, 56, −30, −88, −63, 21, 85, 68, −13,        −82, −74, 4, 78, 78, 4, −74, −82, −13, 68, 85, 21, −63, −88,        −30, 56, 89, 38, −50, −90, −45, 42, 90, 53, −34, −88, −60, 26,        86, 66, −17, −84, −72, 9, 80, 77}    -   {78, 77, −4, −80, −74, 9, 82, 72, −13, −84, −68, 17, 85, 66,        −21, −86, −63, 26, 88, 60, −30, −88, −56, 34, 89, 53, −38, −90,        −50, 42, 90, 45, −45, −90, −42, 50, 90, 38, −53, −89, −34, 56,        88, 30, −60, −88, −26, 63, 86, 21, −66, −85, −17, 68, 84, 13,        −72, −82, −9, 74, 80, 4, −77, −78}    -   {80, 72, −17, −86, −60, 34, 90, 45, −50, −89, −30, 63, 85, 13,        −74, −78, 4, 82, 68, −21, −88, −56, 38, 90, 42, −53, −88, −26,        66, 84, 9, −77, −77, 9, 84, 66, −26, −88, −53, 42, 90, 38, −56,        −88, −21, 68, 82, 4, −78, −74, 13, 85, 63, −30, −89, −50, 45,        90, 34, −60, −86, −17, 72, 80}    -   {82, 66, −30, −90, −42, 56, 86, 13, −77, −74, 17, 88, 53, −45,        −89, −26, 68, 80, −4, −84, −63, 34, 90, 38, −60, −85, −9, 78,        72, −21, −88, −50, 50, 88, 21, −72, −78, 9, 85, 60, −38, −90,        −34, 63, 84, 4, −80, −68, 26, 89, 45, −53, −88, −17, 74, 77,        −13, −86, −56, 42, 90, 30, −66, −82}    -   {84, 60, −42, −89, −21, 74, 74, −21, −89, −42, 60, 84, 0, −84,        −60, 42, 89, 21, −74, −74, 21, 89, 42, −60, −84, 0, 84, 60, −42,        −89, −21, 74, 74, −21, −89, −42, 60, 84, 0, −84, −60, 42, 89,        21, −74, −74, 21, 89, 42, −60, −84, 0, 84, 60, −42, −89, −21,        74, 74, −21, −89, −42, 60, 84}    -   {85, 53, −53, −85, 0, 85, 53, −53, −85, 0, 85, 53, −53, −85, 0,        85, 53, −53, −85, 0, 85, 53, −53, −85, 0, 85, 53, −53, −85, 0,        85, 53, −53, −85, 0, 85, 53, −53, −85, 0, 85, 53, −53, −85, 0,        85, 53, −53, −85, 0, 85, 53, −53, −85, 0, 85, 53, −53, −85, 0,        85, 53, −53, −85}    -   {86, 45, −63, −78, 21, 90, 26, −77, −66, 42, 88, 4, −85, −50,        60, 80, −17, −90, −30, 74, 68, −38, −88, −9, 84, 53, −56, −82,        13, 89, 34, −72, −72, 34, 89, 13, −82, −56, 53, 84, −9, −88,        −38, 68, 74, −30, −90, −17, 80, 60, −50, −85, 4, 88, 42, −66,        −77, 26, 90, 21, −78, −63, 45, 86}    -   {88, 38, −72, −68, 42, 86, −4, −88, −34, 74, 66, −45, −85, 9,        89, 30, −77, −63, 50, 84, −13, −90, −26, 78, 60, −53, −82, 17,        90, 21, −80, −56, 56, 80, −21, −90, −17, 82, 53, −60, −78, 26,        90, 13, −84, −50, 63, 77, −30, −89, −9, 85, 45, −66, −74, 34,        88, 4, −86, −42, 68, 72, −38, −88}    -   {88, 30, −78, −56, 60, 77, −34, −88, 4, 89, 26, −80, −53, 63,        74, −38, −86, 9, 90, 21, −82, −50, 66, 72, −42, −85, 13, 90, 17,        −84, −45, 68, 68, −45, −84, 17, 90, 13, −85, −42, 72, 66, −50,        −82, 21, 90, 9, −86, −38, 74, 63, −53, −80, 26, 89, 4, −88, −34,        77, 60, −56, −78, 30, 88}    -   {89, 21, −84, −42, 74, 60, −60, −74, 42, 84, −21, −89, 0, 89,        21, −84, −42, 74, 60, −60, −74, 42, 84, −21, −89, 0, 89, 21,        −84, −42, 74, 60, −60, −74, 42, 84, −21, −89, 0, 89, 21, −84,        −42, 74, 60, −60, −74, 42, 84, −21, −89, 0, 89, 21, −84, −42,        74, 60, −60, −74, 42, 84, −21, −89}    -   {90, 13, −88, −26, 84, 38, −78, −50, 72, 60, −63, −68, 53, 77,        −42, −82, 30, 86, −17, −89, 4, 90, 9, −88, −21, 85, 34, −80,        −45, 74, 56, −66, −66, 56, 74, −45, −80, 34, 85, −21, −88, 9,        90, 4, −89, −17, 86, 30, −82, −42, 77, 53, −68, −63, 60, 72,        −50, −78, 38, 84, −26, −88, 13, 90}    -   {90, 4, −90, −9, 89, 13, −88, −17, 88, 21, −86, −26, 85, 30,        −84, −34, 82, 38, −80, −42, 78, 45, −77, −50, 74, 53, −72, −56,        68, 60, −66, −63, 63, 66, −60, −68, 56, 72, −53, −74, 50, 77,        −45, −78, 42, 80, −38, −82, 34, 84, −30, −85, 26, 86, −21, −88,        17, 88, −13, −89, 9, 90, −4, −90}    -   {90, −4, −90, 9, 89, −13, −88, 17, 88, −21, −86, 26, 85, −30,        −84, 34, 82, −38, −80, 42, 78, −45, −77, 50, 74, −53, −72, 56,        68, −60, −66, 63, 63, −66, −60, 68, 56, −72, −53, 74, 50, −77,        −45, 78, 42, −80, −38, 82, 34, −84, −30, 85, 26, −86, −21, 88,        17, −88, −13, 89, 9, −90, −4, 90}    -   {90, −13, −88, 26, 84, −38, −78, 50, 72, −60, −63, 68, 53, −77,        −42, 82, 30, −86, −17, 89, 4, −90, 9, 88, −21, −85, 34, 80, −45,        −74, 56, 66, −66, −56, 74, 45, −80, −34, 85, 21, −88, −9, 90,        −4, −89, 17, 86, −30, −82, 42, 77, −53, −68, 63, 60, −72, −50,        78, 38, −84, −26, 88, 13, −90}    -   {89, −21, −84, 42, 74, −60, −60, 74, 42, −84, −21, 89, 0, −89,        21, 84, −42, −74, 60, 60, −74, −42, 84, 21, −89, 0, 89, −21,        −84, 42, 74, −60, −60, 74, 42, −84, −21, 89, 0, −89, 21, 84,        −42, −74, 60, 60, −74, −42, 84, 21, −89, 0, 89, −21, −84, 42,        74, −60, −60, 74, 42, −84, −21, 89}    -   {88, −30, −78, 56, 60, −77, −34, 88, 4, −89, 26, 80, −53, −63,        74, 38, −86, −9, 90, −21, −82, 50, 66, −72, −42, 85, 13, −90,        17, 84, −45, −68, 68, 45, −84, −17, 90, −13, −85, 42, 72, −66,        −50, 82, 21, −90, 9, 86, −38, −74, 63, 53, −80, −26, 89, −4,        −88, 34, 77, −60, −56, 78, 30, −88}    -   {88, −38, −72, 68, 42, −86, −4, 88, −34, −74, 66, 45, −85, −9,        89, −30, −77, 63, 50, −84, −13, 90, −26, −78, 60, 53, −82, −17,        90, −21, −80, 56, 56, −80, −21, 90, −17, −82, 53, 60, −78, −26,        90, −13, −84, 50, 63, −77, −30, 89, −9, −85, 45, 66, −74, −34,        88, −4, −86, 42, 68, −72, −38, 88}    -   {86, −45, −63, 78, 21, −90, 26, 77, −66, −42, 88, −4, −85, 50,        60, −80, −17, 90, −30, −74, 68, 38, −88, 9, 84, −53, −56, 82,        13, −89, 34, 72, −72, −34, 89, −13, −82, 56, 53, −84, −9, 88,        −38, −68, 74, 30, −90, 17, 80, −60, −50, 85, 4, −88, 42, 66,        −77, −26, 90, −21, −78, 63, 45, −86}    -   {85, −53, −53, 85, 0, −85, 53, 53, −85, 0, 85, −53, −53, 85, 0,        −85, 53, 53, −85, 0, 85, −53, −53, 85, 0, −85, 53, 53, −85, 0,        85, −53, −53, 85, 0, −85, 53, 53, −85, 0, 85, −53, −53, 85, 0,        −85, 53, 53, −85, 0, 85, −53, −53, 85, 0, −85, 53, 53, −85, 0,        85, −53, −53, 85}    -   {84, −60, −42, 89, −21, −74, 74, 21, −89, 42, 60, −84, 0, 84,        −60, −42, 89, −21, −74, 74, 21, −89, 42, 60, −84, 0, 84, −60,        −42, 89, −21, −74, 74, 21, −89, 42, 60, −84, 0, 84, −60, −42,        89, −21, −74, 74, 21, −89, 42, 60, −84, 0, 84, −60, −42, 89,        −21, −74, 74, 21, −89, 42, 60, −84}    -   {82, −66, −30, 90, −42, −56, 86, −13, −77, 74, 17, −88, 53, 45,        −89, 26, 68, −80, −4, 84, −63, −34, 90, −38, −60, 85, −9, −78,        72, 21, −88, 50, 50, −88, 21, 72, −78, −9, 85, −60, −38, 90,        −34, −63, 84, −4, −80, 68, 26, −89, 45, 53, −88, 17, 74, −77,        −13, 86, −56, −42, 90, −30, −66, 82}    -   {80, −72, −17, 86, −60, −34, 90, −45, −50, 89, −30, −63, 85,        −13, −74, 78, 4, −82, 68, 21, −88, 56, 38, −90, 42, 53, −88, 26,        66, −84, 9, 77, −77, −9, 84, −66, −26, 88, −53, −42, 90, −38,        −56, 88, −21, −68, 82, −4, −78, 74, 13, −85, 63, 30, −89, 50,        45, −90, 34, 60, −86, 17, 72, −80}    -   {78, −77, −4, 80, −74, −9, 82, −72, −13, 84, −68, −17, 85, −66,        −21, 86, −63, −26, 88, −60, −30, 88, −56, −34, 89, −53, −38, 90,        −50, −42, 90, −45, −45, 90, −42, −50, 90, −38, −53, 89, −34,        −56, 88, −30, −60, 88, −26, −63, 86, −21, −66, 85, −17, −68, 84,        −13, −72, 82, −9, −74, 80, −4, −77, 78}    -   {77, −80, 9, 72, −84, 17, 66, −86, 26, 60, −88, 34, 53, −90, 42,        45, −90, 50, 38, −89, 56, 30, −88, 63, 21, −85, 68, 13, −82, 74,        4, −78, 78, −4, −74, 82, −13, −68, 85, −21, −63, 88, −30, −56,        89, −38, −50, 90, −45, −42, 90, −53, −34, 88, −60, −26, 86, −66,        −17, 84, −72, −9, 80, −77}    -   {74, −84, 21, 60, −89, 42, 42, −89, 60, 21, −84, 74, 0, −74, 84,        −21, −60, 89, −42, −42, 89, −60, −21, 84, −74, 0, 74, −84, 21,        60, −89, 42, 42, −89, 60, 21, −84, 74, 0, −74, 84, −21, −60, 89,        −42, −42, 89, −60, −21, 84, −74, 0, 74, −84, 21, 60, −89, 42,        42, −89, 60, 21, −84, 74}    -   {72, −86, 34, 45, −89, 63, 13, −78, 82, −21, −56, 90, −53, −26,        84, −77, 9, 66, −88, 42, 38, −88, 68, 4, −74, 85, −30, −50, 90,        −60, −17, 80, −80, 17, 60, −90, 50, 30, −85, 74, −4, −68, 88,        −38, −42, 88, −66, −9, 77, −84, 26, 53, −90, 56, 21, −82, 78,        −13, −63, 89, −45, −34, 86, −72}    -   {68, −88, 45, 30, −84, 78, −17, −56, 90, −60, −13, 77, −85, 34,        42, −88, 72, −4, −66, 89, −50, −26, 82, −80, 21, 53, −90, 63, 9,        −74, 86, −38, −38, 86, −74, 9, 63, −90, 53, 21, −80, 82, −26,        −50, 89, −66, −4, 72, −88, 42, 34, −85, 77, −13, −60, 90, −56,        −17, 78, −84, 30, 45, −88, 68}    -   {66, −90, 56, 13, −74, 88, −45, −26, 80, −84, 34, 38, −85, 78,        −21, −50, 88, −72, 9, 60, −90, 63, 4, −68, 89, −53, −17, 77,        −86, 42, 30, −82, 82, −30, −42, 86, −77, 17, 53, −89, 68, −4,        −63, 90, −60, −9, 72, −88, 50, 21, −78, 85, −38, −34, 84, −80,        26, 45, −88, 74, −13, −56, 90, −66}    -   {63, −90, 66, −4, −60, 90, −68, 9, 56, −89, 72, −13, −53, 88,        −74, 17, 50, −88, 77, −21, −45, 86, −78, 26, 42, −85, 80, −30,        −38, 84, −82, 34, 34, −82, 84, −38, −30, 80, −85, 42, 26, −78,        86, −45, −21, 77, −88, 50, 17, −74, 88, −53, −13, 72, −89, 56,        9, −68, 90, −60, −4, 66, −90, 63}    -   {60, −89, 74, −21, −42, 84, −84, 42, 21, −74, 89, −60, 0, 60,        −89, 74, −21, −42, 84, −84, 42, 21, −74, 89, −60, 0, 60, −89,        74, −21, −42, 84, −84, 42, 21, −74, 89, −60, 0, 60, −89, 74,        −21, −42, 84, −84, 42, 21, −74, 89, −60, 0, 60, −89, 74, −21,        −42, 84, −84, 42, 21, −74, 89, −60}    -   {56, −88, 80, −38, −21, 72, −90, 68, −17, −42, 82, −86, 53, 4,        −60, 88, −78, 34, 26, −74, 90, −66, 13, 45, −84, 85, −50, −9,        63, −89, 77, −30, −30, 77, −89, 63, −9, −50, 85, −84, 45, 13,        −66, 90, −74, 26, 34, −78, 88, −60, 4, 53, −86, 82, −42, −17,        68, −90, 72, −21, −38, 80, −88, 56}    -   {53, −85, 85, −53, 0, 53, −85, 85, −53, 0, 53, −85, 85, −53, 0,        53, −85, 85, −53, 0, 53, −85, 85, −53, 0, 53, −85, 85, −53, 0,        53, −85, 85, −53, 0, 53, −85, 85, −53, 0, 53, −85, 85, −53, 0,        53, −85, 85, −53, 0, 53, −85, 85, −53, 0, 53, −85, 85, −53, 0,        53, −85, 85, −53}    -   {50, −82, 88, −66, 21, 30, −72, 90, −78, 42, 9, −56, 85, −86,        60, −13, −38, 77, −90, 74, −34, −17, 63, −88, 84, −53, 4, 45,        −80, 89, −68, 26, 26, −68, 89, −80, 45, 4, −53, 84, −88, 63,        −17, −34, 74, −90, 77, −38, −13, 60, −86, 85, −56, 9, 42, −78,        90, −72, 30, 21, −66, 88, −82, 50}    -   {45, −78, 90, −77, 42, 4, −50, 80, −90, 74, −38, −9, 53, −82,        89, −72, 34, 13, −56, 84, −88, 68, −30, −17, 60, −85, 88, −66,        26, 21, −63, 86, −86, 63, −21, −26, 66, −88, 85, −60, 17, 30,        −68, 88, −84, 56, −13, −34, 72, −89, 82, −53, 9, 38, −74, 90,        −80, 50, −4, −42, 77, −90, 78, −45}    -   {42, −74, 89, −84, 60, −21, −21, 60, −84, 89, −74, 42, 0, −42,        74, −89, 84, −60, 21, 21, −60, 84, −89, 74, −42, 0, 42, −74, 89,        −84, 60, −21, −21, 60, −84, 89, −74, 42, 0, −42, 74, −89, 84,        −60, 21, 21, −60, 84, −89, 74, −42, 0, 42, −74, 89, −84, 60,        −21, −21, 60, −84, 89, −74, 42}    -   {38, −68, 86, −88, 74, −45, 9, 30, −63, 84, −90, 78, −53, 17,        21, −56, 80, −90, 82, −60, 26, 13, −50, 77, −89, 85, −66, 34, 4,        −42, 72, −88, 88, −72, 42, −4, −34, 66, −85, 89, −77, 50, −13,        −26, 60, −82, 90, −80, 56, −21, −17, 53, −78, 90, −84, 63, −30,        −9, 45, −74, 88, −86, 68, −38}    -   {34, −63, 82, −90, 84, −66, 38, −4, −30, 60, −80, 90, −85, 68,        −42, 9, 26, −56, 78, −89, 86, −72, 45, −13, −21, 53, −77, 88,        −88, 74, −50, 17, 17, −50, 74, −88, 88, −77, 53, −21, −13, 45,        −72, 86, −89, 78, −56, 26, 9, −42, 68, −85, 90, −80, 60, −30,        −4, 38, −66, 84, −90, 82, −63, 34}    -   {30, −56, 77, −88, 89, −80, 63, −38, 9, 21, −50, 72, −85, 90,        −84, 68, −45, 17, 13, −42, 66, −82, 90, −86, 74, −53, 26, 4,        −34, 60, −78, 88, −88, 78, −60, 34, −4, −26, 53, −74, 86, −90,        82, −66, 42, −13, −17, 45, −68, 84, −90, 85, −72, 50, −21, −9,        38, −63, 80, −89, 88, −77, 56, −30}    -   {26, −50, 68, −82, 89, −88, 80, −66, 45, −21, −4, 30, −53, 72,        −84, 90, −88, 78, −63, 42, −17, −9, 34, −56, 74, −85, 90, −86,        77, −60, 38, −13, −13, 38, −60, 77, −86, 90, −85, 74, −56, 34,        −9, −17, 42, −63, 78, −88, 90, −84, 72, −53, 30, −4, −21, 45,        −66, 80, −88, 89, −82, 68, −50, 26}    -   {21, −42, 60, −74, 84, −89, 89, −84, 74, −60, 42, −21, 0, 21,        −42, 60, −74, 84, −89, 89, −84, 74, −60, 42, −21, 0, 21, −42,        60, −74, 84, −89, 89, −84, 74, −60, 42, −21, 0, 21, −42, 60,        −74, 84, −89, 89, −84, 74, −60, 42, −21, 0, 21, −42, 60, −74,        84, −89, 89, −84, 74, −60, 42, −21}    -   {17, −34, 50, −63, 74, −82, 88, −90, 88, −84, 77, −66, 53, −38,        21, −4, −13, 30, −45, 60, −72, 80, −86, 90, −89, 85, −78, 68,        −56, 42, −26, 9, 9, −26, 42, −56, 68, −78, 85, −89, 90, −86, 80,        −72, 60, −45, 30, −13, −4, 21, −38, 53, −66, 77, −84, 88, −90,        88, −82, 74, −63, 50, −34, 17}    -   {13, −26, 38, −50, 60, −68, 77, −82, 86, −89, 90, −88, 85, −80,        74, −66, 56, −45, 34, −21, 9, 4, −17, 30, −42, 53, −63, 72, −78,        84, −88, 90, −90, 88, −84, 78, −72, 63, −53, 42, −30, 17, −4,        −9, 21, −34, 45, −56, 66, −74, 80, −85, 88, −90, 89, −86, 82,        −77, 68, −60, 50, −38, 26, −13}    -   {9, −17, 26, −34, 42, −50, 56, −63, 68, −74, 78, −82, 85, −88,        89, −90, 90, −88, 86, −84, 80, −77, 72, −66, 60, −53, 45, −38,        30, −21, 13, −4, −4, 13, −21, 30, −38, 45, −53, 60, −66, 72,        −77, 80, −84, 86, −88, 90, −90, 89, −88, 85, −82, 78, −74, 68,        −63, 56, −50, 42, −34, 26, −17, 9}    -   {4, −9, 13, −17, 21, −26, 30, −34, 38, −42, 45, −50, 53, −56,        60, −63, 66, −68, 72, −74, 77, −78, 80, −82, 84, −85, 86, −88,        88, −89, 90, −90, 90, −90, 89, −88, 88, −86, 85, −84, 82, −80,        78, −77, 74, −72, 68, −66, 63, −60, 56, −53, 50, −45, 42, −38,        34, −30, 26, −21, 17, −13, 9, −4}.

In an embodiment, one or more 8-bit DCT-5 transform cores may be usedfor the primary transform. As an example, the 8-bit DCT-5 transformcores may be:

4-Point DCT-5:

-   -   {47, 69, 69, 69}    -   {69, 61, −21, −87}    -   {69, −21, −87, 61}    -   {69, −87, 61, −21}

8-Point DCT-5:

-   -   {47, 66, 66, 66, 66, 66, 66, 66}    -   {66, 86, 62, 29, −10, −47, −76, −91}    -   {66, 62, −10, −76, −91, −47, 29, 86}    -   {66, 29, −76, −76, 29, 93, 29, −76}    -   {66, −10, −91, 29, 86, −47, −76, 62}    -   {66, −47, −47, 93, −47, −47, 93, −47}    -   {66, −76, 29, 29, −76, 93, −76, 29}    -   {66, −91, 86, −76, 62, −47, 29, −10}

16-Point DCT-5:

-   -   {45, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65}    -   {65, 89, 85, 75, 64, 48, 33, 15, −5, −23, −39, −56, −71, −81,        −88, −91}    -   {65, 85, 64, 33, −5, −39, −71, −88, −91, −81, −56, −23, 15, 48,        75, 89}    -   {65, 75, 33, −23, −71, −91, −81, −39, 15, 64, 89, 85, 48, −5,        −56, −88}    -   {65, 64, −5, −71, −91, −56, 15, 75, 89, 48, −23, −81, −88, −39,        33, 85}    -   {65, 48, −39, −91, −56, 33, 89, 64, −23, −88, −71, 15, 85, 75,        −5, −81}    -   {65, 33, −71, −81, 15, 89, 48, −56, −88, −5, 85, 64, −39, −91,        −23, 75}    -   {65, 15, −88, −39, 75, 64, −56, −81, 33, 89, −5, −91, −23, 85,        48, −71}    -   {65, −5, −91, 15, 89, −23, −88, 33, 85, −39, −81, 48, 75, −56,        −71, 64}    -   {65, −23, −81, 64, 48, −88, −5, 89, −39, −71, 75, 33, −91, 15,        85, −56}    -   {65, −39, −56, 89, −23, −71, 85, −5, −81, 75, 15, −88, 64, 33,        −91, 48}    -   {65, −56, −23, 85, −81, 15, 64, −91, 48, 33, −88, 75, −5, −71,        89, −39}    -   {65, −71, 15, 48, −88, 85, −39, −23, 75, −91, 64, −5, −56, 89,        −81, 33}    -   {65, −81, 48, −5, −39, 75, −91, 85, −56, 15, 33, −71, 89, −88,        64, −23}    -   {65, −88, 75, −56, 33, −5, −23, 48, −71, 85, −91, 89, −81, 64,        −39, 15}    -   {65, −91, 89, −88, 85, −81, 75, −71, 64, −56, 48, −39, 33, −23,        15, −5}

32-Point DCT-5:

-   -   {46, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65,        65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65}    -   {65, 91, 89, 87, 84, 80, 75, 70, 64, 57, 49, 42, 33, 25, 16, 7,        −2, −12, −20, −29, −38, −46, −53, −60, −67, −72, −78, −82, −86,        −89, −90, −91}    -   {65, 89, 84, 75, 64, 49, 33, 16, −2, −20, −38, −53, −67, −78,        −86, −90, −91, −89, −82, −72, −60, −46, −29, −12, 7, 25, 42, 57,        70, 80, 87, 91}    -   {65, 87, 75, 57, 33, 7, −20, −46, −67, −82, −90, −90, −82, −67,        −46, −20, 7, 33, 57, 75, 87, 92, 87, 75, 57, 33, 7, −20, −46,        −67, −82, −90}    -   {65, 84, 64, 33, −2, −38, −67, −86, −91, −82, −60, −29, 7, 42,        70, 87, 91, 80, 57, 25, −12, −46, −72, −89, −90, −78, −53, −20,        16, 49, 75, 89}    -   {65, 80, 49, 7, −38, −72, −90, −86, −60, −20, 25, 64, 87, 89,        70, 33, −12, −53, −82, −91, −78, −46, −2, 42, 75, 91, 84, 57,        16, −29, −67, −89}    -   {65, 75, 33, −20, −67, −90, −82, −46, 7, 57, 87, 87, 57, 7, −46,        −82, −90, −67, −20, 33, 75, 92, 75, 33, −20, −67, −90, −82, −46,        7, 57, 87}    -   {65, 70, 16, −46, −86, −86, −46, 16, 70, 92, 70, 16, −46, −86,        −86, −46, 16, 70, 92, 70, 16, −46, −86, −86, −46, 16, 70, 92,        70, 16, −46, −86}    -   {65, 64, −2, −67, −91, −60, 7, 70, 91, 57, −12, −72, −90, −53,        16, 75, 89, 49, −20, −78, −89, −46, 25, 80, 87, 42, −29, −82,        −86, −38, 33, 84}    -   {65, 57, −20, −82, −82, −20, 57, 92, 57, −20, −82, −82, −20, 57,        92, 57, −20, −82, −82, −20, 57, 92, 57, −20, −82, −82, −20, 57,        92, 57, −20, −82}    -   {65, 49, −38, −90, −60, 25, 87, 70, −12, −82, −78, −2, 75, 84,        16, −67, −89, −29, 57, 91, 42, −46, −91, −53, 33, 89, 64, −20,        −86, −72, 7, 80}    -   {65, 42, −53, −90, −29, 64, 87, 16, −72, −82, −2, 80, 75, −12,        −86, −67, 25, 89, 57, −38, −91, −46, 49, 91, 33, −60, −89, −20,        70, 84, 7, −78}    -   {65, 33, −67, −82, 7, 87, 57, −46, −90, −20, 75, 75, −20, −90,        −46, 57, 87, 7, −82, −67, 33, 92, 33, −67, −82, 7, 87, 57, −46,        −90, −20, 75}    -   {65, 25, −78, −67, 42, 89, 7, −86, −53, 57, 84, −12, −90, −38,        70, 75, −29, −91, −20, 80, 64, −46, −89, −2, 87, 49, −60, −82,        16, 91, 33, −72}    -   {65, 16, −86, −46, 70, 70, −46, −86, 16, 92, 16, −86, −46, 70,        70, −46, −86, 16, 92, 16, −86, −46, 70, 70, −46, −86, 16, 92,        16, −86, −46, 70}    -   {65, 7, −90, −20, 87, 33, −82, −46, 75, 57, −67, −67, 57, 75,        −46, −82, 33, 87, −20, −90, 7, 92, 7, −90, −20, 87, 33, −82,        −46, 75, 57, −67}    -   {65, −2, −91, 7, 91, −12, −90, 16, 89, −20, −89, 25, 87, −29,        −86, 33, 84, −38, −82, 42, 80, −46, −78, 49, 75, −53, −72, 57,        70, −60, −67, 64}    -   {65, −12, −89, 33, 80, −53, −67, 70, 49, −82, −29, 89, 7, −91,        16, 87, −38, −78, 57, 64, −72, −46, 84, 25, −90, −2, 91, −20,        −86, 42, 75, −60}    -   {65, −20, −82, 57, 57, −82, −20, 92, −20, −82, 57, 57, −82, −20,        92, −20, −82, 57, 57, −82, −20, 92, −20, −82, 57, 57, −82, −20,        92, −20, −82, 57}    -   {65, −29, −72, 75, 25, −91, 33, 70, −78, −20, 91, −38, −67, 80,        16, −90, 42, 64, −82, −12, 89, −46, −60, 84, 7, −89, 49, 57,        −86, −2, 87, −53}    -   {65, −38, −60, 87, −12, −78, 75, 16, −89, 57, 42, −91, 33, 64,        −86, 7, 80, −72, −20, 89, −53, −46, 91, −29, −67, 84, −2, −82,        70, 25, −90, 49}    -   {65, −46, −46, 92, −46, −46, 92, −46, −46, 92, −46, −46, 92,        −46, −46, 92, −46, −46, 92, −46, −46, 92, −46, −46, 92, −46,        −46, 92, −46, −46, 92, −46}    -   {65, −53, −29, 87, −72, −2, 75, −86, 25, 57, −91, 49, 33, −89,        70, 7, −78, 84, −20, −60, 91, −46, −38, 89, −67, −12, 80, −82,        16, 64, −90, 42}    -   {65, −60, −12, 75, −89, 42, 33, −86, 80, −20, −53, 91, −67, −2,        70, −90, 49, 25, −82, 84, −29, −46, 89, −72, 7, 64, −91, 57, 16,        −78, 87, −38}    -   {65, −67, 7, 57, −90, 75, −20, −46, 87, −82, 33, 33, −82, 87,        −46, −20, 75, −90, 57, 7, −67, 92, −67, 7, 57, −90, 75, −20,        −46, 87, −82, 33}    -   {65, −72, 25, 33, −78, 91, −67, 16, 42, −82, 89, −60, 7, 49,        −86, 87, −53, −2, 57, −89, 84, −46, −12, 64, −90, 80, −38, −20,        70, −91, 75, −29}    -   {65, −78, 42, 7, −53, 84, −90, 70, −29, −20, 64, −89, 87, −60,        16, 33, −72, 91, −82, 49, −2, −46, 80, −91, 75, −38, −12, 57,        −86, 89, −67, 25}    -   {65, −82, 57, −20, −20, 57, −82, 92, −82, 57, −20, −20, 57, −82,        92, −82, 57, −20, −20, 57, −82, 92, −82, 57, −20, −20, 57, −82,        92, −82, 57, −20}    -   {65, −86, 70, −46, 16, 16, −46, 70, −86, 92, −86, 70, −46, 16,        16, −46, 70, −86, 92, −86, 70, −46, 16, 16, −46, 70, −86, 92,        −86, 70, −46, 16}    -   {65, −89, 80, −67, 49, −29, 7, 16, −38, 57, −72, 84, −90, 91,        −86, 75, −60, 42, −20, −2, 25, −46, 64, −78, 87, −91, 89, −82,        70, −53, 33, −12}    -   {65, −90, 87, −82, 75, −67, 57, −46, 33, −20, 7, 7, −20, 33,        −46, 57, −67, 75, −82, 87, −90, 92, −90, 87, −82, 75, −67, 57,        −46, 33, −20, 7}    -   {65, −91, 91, −90, 89, −89, 87, −86, 84, −82, 80, −78, 75, −72,        70, −67, 64, −60, 57, −53, 49, −46, 42, −38, 33, −29, 25, −20,        16, −12, 7, −2}

64-Point DCT-5:

-   -   {44, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,        64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,        64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,        64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64}    -   {64, 91, 90, 90, 90, 88, 87, 85, 84, 82, 80, 78, 75, 72, 70, 66,        64, 61, 58, 54, 50, 46, 42, 39, 34, 29, 25, 22, 17, 13, 7, 4,        −1, −6, −10, −15, −19, −24, −28, −32, −36, −40, −44, −48, −51,        −55, −59, −63, −65, −68, −70, −73, −77, −79, −81, −83, −85, −86,        −88, −89, −90, −90, −91, −91}    -   {64, 90, 90, 87, 84, 80, 75, 70, 64, 58, 50, 42, 34, 25, 17, 7,        −1, −10, −19, −28, −36, −44, −51, −59, −65, −70, −77, −81, −85,        −88, −90, −91, −91, −90, −89, −86, −83, −79, −73, −68, −63, −55,        −48, −40, −32, −24, −15, −6, 4, 13, 22, 29, 39, 46, 54, 61, 66,        72, 78, 82, 85, 88, 90, 91}    -   {64, 90, 87, 82, 75, 66, 58, 46, 34, 22, 7, −6, −19, −32, −44,        −55, −65, −73, −81, −86, −90, −91, −90, −88, −83, −77, −68, −59,        −48, −36, −24, −10, 4, 17, 29, 42, 54, 64, 72, 80, 85, 90, 91,        90, 88, 84, 78, 70, 61, 50, 39, 25, 13, −1, −15, −28, −40, −51,        −63, −70, −79, −85, −89, −91}    -   {64, 90, 84, 75, 64, 50, 34, 17, −1, −19, −36, −51, −65, −77,        −85, −90, −91, −89, −83, −73, −63, −48, −32, −15, 4, 22, 39, 54,        66, 78, 85, 90, 91, 88, 82, 72, 61, 46, 29, 13, −6, −24, −40,        −55, −68, −79, −86, −90, −91, −88, −81, −70, −59, −44, −28, −10,        7, 25, 42, 58, 70, 80, 87, 90}    -   {64, 88, 80, 66, 50, 29, 7, −15, −36, −55, −70, −83, −90, −91,        −86, −77, −63, −44, −24, −1, 22, 42, 61, 75, 85, 90, 90, 84, 72,        58, 39, 17, −6, −28, −48, −65, −79, −88, −91, −89, −81, −68,        −51, −32, −10, 13, 34, 54, 70, 82, 90, 91, 87, 78, 64, 46, 25,        4, −19, −40, −59, −73, −85, −90}    -   {64, 87, 75, 58, 34, 7, −19, −44, −65, −81, −90, −90, −83, −68,        −48, −24, 4, 29, 54, 72, 85, 91, 88, 78, 61, 39, 13, −15, −40,        −63, −79, −89, −91, −85, −70, −51, −28, −1, 25, 50, 70, 84, 90,        90, 80, 64, 42, 17, −10, −36, −59, −77, −88, −91, −86, −73, −55,        −32, −6, 22, 46, 66, 82, 90}    -   {64, 85, 70, 46, 17, −15, −44, −68, −85, −91, −86, −70, −48,        −19, 13, 42, 66, 84, 91, 87, 72, 50, 22, −10, −40, −65, −83,        −91, −88, −73, −51, −24, 7, 39, 64, 82, 90, 88, 75, 54, 25, −6,        −36, −63, −81, −90, −89, −77, −55, −28, 4, 34, 61, 80, 90, 90,        78, 58, 29, −1, −32, −59, −79, −90}    -   {64, 84, 64, 34, −1, −36, −65, −85, −91, −83, −63, −32, 4, 39,        66, 85, 91, 82, 61, 29, −6, −40, −68, −86, −91, −81, −59, −28,        7, 42, 70, 87, 90, 80, 58, 25, −10, −44, −70, −88, −90, −79,        −55, −24, 13, 46, 72, 88, 90, 78, 54, 22, −15, −48, −73, −89,        −90, −77, −51, −19, 17, 50, 75, 90}    -   {64, 82, 58, 22, −19, −55, −81, −91, −83, −59, −24, 17, 54, 80,        91, 84, 61, 25, −15, −51, −79, −91, −85, −63, −28, 13, 50, 78,        90, 85, 64, 29, −10, −48, −77, −90, −86, −65, −32, 7, 46, 75,        90, 87, 66, 34, −6, −44, −73, −90, −88, −68, −36, 4, 42, 72, 90,        88, 70, 39, −1, −40, −70, −89}    -   {64, 80, 50, 7, −36, −70, −90, −86, −63, −24, 22, 61, 85, 90,        72, 39, −6, −48, −79, −91, −81, −51, −10, 34, 70, 90, 87, 64,        25, −19, −59, −85, −90, −73, −40, 4, 46, 78, 91, 82, 54, 13,        −32, −68, −89, −88, −65, −28, 17, 58, 84, 90, 75, 42, −1, −44,        −77, −91, −83, −55, −15, 29, 66, 88}    -   {64, 78, 42, −6, −51, −83, −90, −70, −32, 17, 61, 87, 88, 64,        22, −28, −68, −90, −85, −55, −10, 39, 75, 91, 80, 46, −1, −48,        −81, −91, −73, −36, 13, 58, 85, 90, 66, 25, −24, −65, −89, −86,        −59, −15, 34, 72, 90, 82, 50, 4, −44, −79, −91, −77, −40, 7, 54,        84, 90, 70, 29, −19, −63, −88}    -   {64, 75, 34, −19, −65, −90, −83, −48, 4, 54, 85, 88, 61, 13,        −40, −79, −91, −70, −28, 25, 70, 90, 80, 42, −10, −59, −88, −86,        −55, −6, 46, 82, 90, 66, 22, −32, −73, −91, −77, −36, 17, 64,        90, 84, 50, −1, −51, −85, −89, −63, −15, 39, 78, 91, 72, 29,        −24, −68, −90, −81, −44, 7, 58, 87}    -   {64, 72, 25, −32, −77, −91, −68, −19, 39, 80, 90, 64, 13, −44,        −83, −89, −59, −6, 50, 85, 87, 54, −1, −55, −88, −85, −48, 7,        61, 90, 82, 42, −15, −65, −90, −79, −36, 22, 70, 91, 75, 29,        −28, −73, −91, −70, −24, 34, 78, 90, 66, 17, −40, −81, −90, −63,        −10, 46, 84, 88, 58, 4, −51, −86}    -   {64, 70, 17, −44, −85, −86, −48, 13, 66, 91, 72, 22, −40, −83,        −88, −51, 7, 64, 90, 75, 25, −36, −81, −89, −55, 4, 61, 90, 78,        29, −32, −79, −90, −59, −1, 58, 90, 80, 34, −28, −77, −90, −63,        −6, 54, 88, 82, 39, −24, −73, −91, −65, −10, 50, 87, 84, 42,        −19, −70, −91, −68, −15, 46, 85}    -   {64, 66, 7, −55, −90, −77, −24, 42, 85, 84, 39, −28, −79, −89,        −51, 13, 70, 91, 64, 4, −59, −90, −73, −19, 46, 87, 82, 34, −32,        −81, −88, −48, 17, 72, 90, 61, −1, −63, −91, −70, −15, 50, 88,        80, 29, −36, −83, −86, −44, 22, 75, 90, 58, −6, −65, −91, −68,        −10, 54, 90, 78, 25, −40, −85}    -   {64, 64, −1, −65, −91, −63, 4, 66, 91, 61, −6, −68, −91, −59, 7,        70, 90, 58, −10, −70, −90, −55, 13, 72, 90, 54, −15, −73, −90,        −51, 17, 75, 90, 50, −19, −77, −89, −48, 22, 78, 88, 46, −24,        −79, −88, −44, 25, 80, 87, 42, −28, −81, −86, −40, 29, 82, 85,        39, −32, −83, −85, −36, 34, 84}    -   {64, 61, −10, −73, −89, −44, 29, 84, 82, 25, −48, −90, −70, −6,        64, 91, 58, −15, −77, −88, −40, 34, 85, 80, 22, −51, −90, −68,        −1, 66, 90, 54, −19, −79, −86, −36, 39, 87, 78, 17, −55, −91,        −65, 4, 70, 90, 50, −24, −81, −85, −32, 42, 88, 75, 13, −59,        −91, −63, 7, 72, 90, 46, −28, −83}    -   {64, 58, −19, −81, −83, −24, 54, 91, 61, −15, −79, −85, −28, 50,        90, 64, −10, −77, −86, −32, 46, 90, 66, −6, −73, −88, −36, 42,        90, 70, −1, −70, −89, −40, 39, 88, 72, 4, −68, −90, −44, 34, 87,        75, 7, −65, −90, −48, 29, 85, 78, 13, −63, −91, −51, 25, 84, 80,        17, −59, −91, −55, 22, 82}    -   {64, 54, −28, −86, −73, −1, 72, 87, 29, −51, −91, −55, 25, 85,        75, 4, −70, −88, −32, 50, 91, 58, −24, −85, −77, −6, 70, 88, 34,        −48, −91, −59, 22, 84, 78, 7, −68, −89, −36, 46, 90, 61, −19,        −83, −79, −10, 66, 90, 39, −44, −90, −63, 17, 82, 80, 13, −65,        −90, −40, 42, 90, 64, −15, −81}    -   {64, 50, −36, −90, −63, 22, 85, 72, −6, −79, −81, −10, 70, 87,        25, −59, −90, −40, 46, 91, 54, −32, −89, −65, 17, 84, 75, −1,        −77, −83, −15, 66, 88, 29, −55, −91, −44, 42, 90, 58, −28, −88,        −68, 13, 82, 78, 4, −73, −85, −19, 64, 90, 34, −51, −91, −48,        39, 90, 61, −24, −86, −70, 7, 80}    -   {64, 46, −44, −91, −48, 42, 91, 50, −40, −91, −51, 39, 90, 54,        −36, −90, −55, 34, 90, 58, −32, −90, −59, 29, 90, 61, −28, −89,        −63, 25, 88, 64, −24, −88, −65, 22, 87, 66, −19, −86, −68, 17,        85, 70, −15, −85, −70, 13, 84, 72, −10, −83, −73, 7, 82, 75, −6,        −81, −77, 4, 80, 78, −1, −79}    -   {64, 42, −51, −90, −32, 61, 88, 22, −68, −85, −10, 75, 80, −1,        −81, −73, 13, 85, 66, −24, −89, −59, 34, 90, 50, −44, −91, −40,        54, 90, 29, −63, −88, −19, 70, 84, 7, −77, −79, 4, 82, 72, −15,        −86, −65, 25, 90, 58, −36, −91, −48, 46, 91, 39, −55, −90, −28,        64, 87, 17, −70, −83, −6, 78}    -   {64, 39, −59, −88, −15, 75, 78, −10, −86, −63, 34, 91, 42, −55,        −89, −19, 72, 80, −6, −85, −65, 29, 90, 46, −51, −90, −24, 70,        82, −1, −83, −68, 25, 90, 50, −48, −90, −28, 66, 84, 4, −81,        −70, 22, 90, 54, −44, −91, −32, 64, 85, 7, −79, −73, 17, 88, 58,        −40, −91, −36, 61, 87, 13, −77}    -   {64, 34, −65, −83, 4, 85, 61, −40, −91, −28, 70, 80, −10, −88,        −55, 46, 90, 22, −73, −77, 17, 90, 50, −51, −89, −15, 78, 72,        −24, −90, −44, 58, 87, 7, −81, −68, 29, 91, 39, −63, −85, −1,        84, 64, −36, −91, −32, 66, 82, −6, −86, −59, 42, 90, 25, −70,        −79, 13, 88, 54, −48, −90, −19, 75}    -   {64, 29, −70, −77, 22, 90, 39, −65, −81, 13, 90, 46, −59, −85,        4, 87, 54, −51, −88, −6, 84, 61, −44, −90, −15, 80, 66, −36,        −91, −24, 75, 72, −28, −91, −32, 70, 78, −19, −90, −40, 64, 82,        −10, −89, −48, 58, 85, −1, −86, −55, 50, 88, 7, −83, −63, 42,        90, 17, −79, −68, 34, 91, 25, −73}    -   {64, 25, −77, −68, 39, 90, 13, −83, −59, 50, 87, −1, −88, −48,        61, 82, −15, −90, −36, 70, 75, −28, −91, −24, 78, 66, −40, −90,        −10, 84, 58, −51, −86, 4, 88, 46, −63, −81, 17, 90, 34, −70,        −73, 29, 91, 22, −79, −65, 42, 90, 7, −85, −55, 54, 85, −6, −89,        −44, 64, 80, −19, −91, −32, 72}    -   {64, 22, −81, −59, 54, 84, −15, −91, −28, 78, 64, −48, −86, 7,        90, 34, −73, −68, 42, 88, −1, −89, −40, 70, 72, −36, −90, −6,        87, 46, −65, −77, 29, 90, 13, −85, −51, 61, 80, −24, −91, −19,        82, 58, −55, −83, 17, 91, 25, −79, −63, 50, 85, −10, −90, −32,        75, 66, −44, −88, 4, 90, 39, −70}    -   {64, 17, −85, −48, 66, 72, −40, −88, 7, 90, 25, −81, −55, 61,        78, −32, −90, −1, 90, 34, −77, −63, 54, 82, −24, −91, −10, 87,        42, −70, −68, 46, 85, −15, −91, −19, 84, 50, −65, −73, 39, 88,        −6, −90, −28, 80, 58, −59, −79, 29, 90, 4, −89, −36, 75, 64,        −51, −83, 22, 91, 13, −86, −44, 70}    -   {64, 13, −88, −36, 78, 58, −63, −73, 42, 85, −19, −91, −6, 90,        29, −81, −51, 66, 70, −48, −83, 25, 90, −1, −90, −24, 84, 46,        −70, −65, 54, 80, −32, −89, 7, 91, 17, −86, −40, 75, 61, −59,        −77, 39, 87, −15, −91, −10, 88, 34, −79, −55, 64, 72, −44, −85,        22, 90, 4, −90, −28, 82, 50, −68}    -   {64, 7, −90, −24, 85, 39, −79, −51, 70, 64, −59, −73, 46, 82,        −32, −88, 17, 90, −1, −91, −15, 88, 29, −83, −44, 75, 58, −65,        −68, 54, 78, −40, −85, 25, 90, −10, −91, −6, 90, 22, −86, −36,        80, 50, −70, −63, 61, 72, −48, −81, 34, 87, −19, −90, 4, 91, 13,        −89, −28, 84, 42, −77, −55, 66}    -   {64, 4, −91, −10, 90, 17, −89, −24, 87, 29, −85, −36, 82, 42,        −79, −48, 75, 54, −70, −59, 66, 64, −63, −68, 58, 72, −51, −77,        46, 80, −40, −83, 34, 85, −28, −88, 22, 90, −15, −90, 7, 91, −1,        −91, −6, 90, 13, −90, −19, 88, 25, −86, −32, 84, 39, −81, −44,        78, 50, −73, −55, 70, 61, −65}    -   {64, −1, −91, 4, 91, −6, −91, 7, 90, −10, −90, 13, 90, −15, −90,        17, 90, −19, −89, 22, 88, −24, −88, 25, 87, −28, −86, 29, 85,        −32, −85, 34, 84, −36, −83, 39, 82, −40, −81, 42, 80, −44, −79,        46, 78, −48, −77, 50, 75, −51, −73, 54, 72, −55, −70, 58, 70,        −59, −68, 61, 66, −63, −65, 64}    -   {64, −6, −90, 17, 88, −28, −85, 39, 80, −48, −73, 58, 66, −65,        −59, 72, 50, −79, −40, 84, 29, −88, −19, 90, 7, −91, 4, 90, −15,        −89, 25, 85, −36, −81, 46, 75, −55, −68, 64, 61, −70, −51, 78,        42, −83, −32, 87, 22, −90, −10, 91, −1, −91, 13, 90, −24, −86,        34, 82, −44, −77, 54, 70, −63}    -   {64, −10, −89, 29, 82, −48, −70, 64, 58, −77, −40, 85, 22, −90,        −1, 90, −19, −86, 39, 78, −55, −65, 70, 50, −81, −32, 88, 13,        −91, 7, 90, −28, −83, 46, 72, −63, −59, 75, 42, −85, −24, 90, 4,        −91, 17, 87, −36, −79, 54, 66, −68, −51, 80, 34, −88, −15, 91,        −6, −90, 25, 84, −44, −73, 61}    -   {64, −15, −86, 42, 72, −65, −51, 82, 25, −90, 4, 90, −32, −79,        58, 61, −77, −36, 88, 7, −91, 22, 84, −48, −68, 70, 46, −85,        −19, 91, −10, −88, 39, 75, −63, −55, 80, 29, −90, −1, 90, −28,        −81, 54, 64, −73, −40, 87, 13, −91, 17, 85, −44, −70, 66, 50,        −83, −24, 90, −6, −89, 34, 78, −59}    -   {64, −19, −83, 54, 61, −79, −28, 90, −10, −86, 46, 66, −73, −36,        90, −1, −89, 39, 72, −68, −44, 87, 7, −90, 29, 78, −63, −51, 84,        17, −91, 22, 82, −55, −59, 80, 25, −91, 13, 85, −48, −65, 75,        34, −90, 4, 88, −40, −70, 70, 42, −88, −6, 90, −32, −77, 64, 50,        −85, −15, 91, −24, −81, 58}    -   {64, −24, −79, 64, 46, −88, −1, 88, −44, −65, 78, 25, −91, 22,        80, −63, −48, 87, 4, −89, 42, 66, −77, −28, 91, −19, −81, 61,        50, −86, −6, 90, −40, −68, 75, 29, −91, 17, 82, −59, −51, 85, 7,        −90, 39, 70, −73, −32, 90, −15, −83, 58, 54, −85, −10, 90, −36,        −70, 72, 34, −90, 13, 84, −55}    -   {64, −28, −73, 72, 29, −91, 25, 75, −70, −32, 91, −24, −77, 70,        34, −91, 22, 78, −68, −36, 90, −19, −79, 66, 39, −90, 17, 80,        −65, −40, 90, −15, −81, 64, 42, −90, 13, 82, −63, −44, 90, −10,        −83, 61, 46, −89, 7, 84, −59, −48, 88, −6, −85, 58, 50, −88, 4,        85, −55, −51, 87, −1, −86, 54}    -   {64, −32, −68, 80, 13, −89, 50, 54, −88, 7, 82, −65, −36, 91,        −28, −70, 78, 17, −90, 46, 58, −86, 4, 84, −63, −40, 90, −24,        −73, 75, 22, −90, 42, 61, −85, −1, 85, −59, −44, 90, −19, −77,        72, 25, −91, 39, 64, −83, −6, 87, −55, −48, 90, −15, −79, 70,        29, −91, 34, 66, −81, −10, 88, −51}    -   {64, −36, −63, 85, −6, −81, 70, 25, −90, 46, 54, −89, 17, 75,        −77, −15, 88, −55, −44, 90, −28, −68, 82, 4, −85, 64, 34, −91,        39, 61, −86, 7, 80, −70, −24, 90, −48, −51, 90, −19, −73, 78,        13, −88, 58, 42, −91, 29, 66, −83, −1, 84, −65, −32, 91, −40,        −59, 87, −10, −79, 72, 22, −90, 50}    -   {64, −40, −55, 90, −24, −68, 84, −6, −79, 75, 13, −86, 64, 29,        −90, 50, 46, −91, 34, 61, −88, 17, 72, −81, −1, 82, −70, −19,        88, −59, −36, 91, −44, −51, 90, −28, −65, 85, −10, −77, 78, 7,        −85, 66, 25, −90, 54, 42, −91, 39, 58, −89, 22, 70, −83, 4, 80,        −73, −15, 87, −63, −32, 90, −48}    -   {64, −44, −48, 91, −40, −51, 90, −36, −55, 90, −32, −59, 90,        −28, −63, 88, −24, −65, 87, −19, −68, 85, −15, −70, 84, −10,        −73, 82, −6, −77, 80, −1, −79, 78, 4, −81, 75, 7, −83, 72, 13,        −85, 70, 17, −86, 66, 22, −88, 64, 25, −89, 61, 29, −90, 58, 34,        −90, 54, 39, −91, 50, 42, −91, 46}    -   {64, −48, −40, 90, −55, −32, 90, −63, −24, 87, −68, −15, 84,        −73, −6, 80, −79, 4, 75, −83, 13, 70, −86, 22, 64, −89, 29, 58,        −90, 39, 50, −91, 46, 42, −91, 54, 34, −90, 61, 25, −88, 66, 17,        −85, 72, 7, −81, 78, −1, −77, 82, −10, −70, 85, −19, −65, 88,        −28, −59, 90, −36, −51, 91, −44}    -   {64, −51, −32, 88, −68, −10, 80, −81, 13, 66, −89, 34, 50, −91,        54, 29, −88, 70, 7, −79, 82, −15, −65, 90, −36, −48, 91, −55,        −28, 87, −70, −6, 78, −83, 17, 64, −90, 39, 46, −91, 58, 25,        −86, 72, 4, −77, 84, −19, −63, 90, −40, −44, 90, −59, −24, 85,        −73, −1, 75, −85, 22, 61, −90, 42}    -   {64, −55, −24, 84, −79, 13, 64, −90, 46, 34, −88, 72, −1, −70,        88, −36, −44, 90, −65, −10, 78, −85, 25, 54, −91, 58, 22, −83,        80, −15, −63, 90, −48, −32, 87, −73, 4, 70, −89, 39, 42, −90,        66, 7, −77, 85, −28, −51, 91, −59, −19, 82, −81, 17, 61, −91,        50, 29, −86, 75, −6, −68, 90, −40}    -   {64, −59, −15, 78, −86, 34, 42, −89, 72, −6, −65, 90, −51, −24,        82, −83, 25, 50, −90, 66, 4, −70, 90, −44, −32, 85, −79, 17, 58,        −91, 61, 13, −77, 87, −36, −40, 88, −73, 7, 64, −91, 54, 22,        −81, 84, −28, −48, 90, −68, −1, 70, −90, 46, 29, −85, 80, −19,        −55, 91, −63, −10, 75, −88, 39}    -   {64, −63, −6, 70, −90, 54, 17, −77, 88, −44, −28, 82, −85, 34,        39, −86, 80, −24, −48, 90, −73, 13, 58, −91, 66, −1, −65, 91,        −59, −10, 72, −90, 50, 22, −79, 87, −40, −32, 84, −83, 29, 42,        −88, 78, −19, −51, 90, −70, 7, 61, −91, 64, 4, −68, 90, −55,        −15, 75, −89, 46, 25, −81, 85, −36}    -   {64, −65, 4, 61, −91, 70, −10, −55, 90, −73, 17, 50, −89, 78,        −24, −44, 87, −81, 29, 39, −85, 84, −36, −32, 82, −86, 42, 25,        −79, 88, −48, −19, 75, −90, 54, 13, −70, 90, −59, −6, 66, −91,        64, −1, −63, 91, −68, 7, 58, −90, 72, −15, −51, 90, −77, 22, 46,        −88, 80, −28, −40, 85, −83, 34}    -   {64, −68, 13, 50, −88, 82, −36, −28, 78, −90, 58, 4, −63, 90,        −73, 22, 42, −85, 85, −44, −19, 72, −91, 64, −6, −55, 90, −79,        29, 34, −81, 88, −51, −10, 66, −91, 70, −15, −48, 87, −83, 39,        25, −77, 90, −59, −1, 61, −90, 75, −24, −40, 84, −86, 46, 17,        −70, 91, −65, 7, 54, −89, 80, −32}    -   {64, −70, 22, 39, −81, 90, −59, 4, 54, −88, 84, −44, −15, 66,        −91, 75, −28, −32, 78, −90, 64, −10, −48, 85, −86, 50, 7, −63,        90, −79, 34, 25, −73, 91, −68, 17, 42, −83, 88, −55, −1, 58,        −89, 82, −40, −19, 70, −91, 72, −24, −36, 80, −90, 61, −6, −51,        87, −85, 46, 13, −65, 90, −77, 29}    -   {64, −73, 29, 25, −70, 91, −77, 34, 22, −68, 90, −79, 39, 17,        −65, 90, −81, 42, 13, −63, 90, −83, 46, 7, −59, 88, −85, 50, 4,        −55, 87, −86, 54, −1, −51, 85, −88, 58, −6, −48, 84, −89, 61,        −10, −44, 82, −90, 64, −15, −40, 80, −90, 66, −19, −36, 78, −91,        70, −24, −32, 75, −91, 72, −28}    - 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  {64, −85, 66, −40, 7, 25, −55, 78, −90, 90, −77, 54, −24, −10,        42, −68, 85, −91, 84, −65, 39, −6, −28, 58, −79, 90, −89, 75,        −51, 22, 13, −44, 70, −86, 91, −83, 64, −36, 4, 29, −59, 80,        −90, 88, −73, 50, −19, −15, 46, −70, 87, −91, 82, −63, 34, −1,        −32, 61, −81, 90, −88, 72, −48, 17}    -   {64, −86, 72, −51, 25, 4, −32, 58, −77, 88, −91, 84, −68, 46,        −19, −10, 39, −63, 80, −90, 90, −81, 64, −40, 13, 17, −44, 66,        −83, 90, −89, 78, −59, 34, −6, −24, 50, −70, 85, −91, 87, −73,        54, −28, −1, 29, −55, 75, −88, 91, −85, 70, −48, 22, 7, −36, 61,        −79, 90, −90, 82, −65, 42, −15}    -   {64, −88, 78, −63, 42, −19, −6, 29, −51, 70, −83, 90, −90, 84,        −70, 54, −32, 7, 17, −40, 61, −77, 87, −91, 88, −79, 64, −44,        22, 4, −28, 50, −68, 82, −90, 90, −85, 72, −55, 34, −10, −15,        39, −59, 75, −86, 91, −89, 80, −65, 46, −24, −1, 25, −48, 66,        −81, 90, −91, 85, −73, 58, −36, 13}    -   {64, −89, 82, −70, 58, −40, 22, −1, −19, 39, −55, 70, −81, 88,        −91, 90, −83, 72, −59, 42, −24, 4, 17, −36, 54, −68, 80, −88,        91, −90, 84, −73, 61, −44, 25, −6, −15, 34, −51, 66, −79, 87,        −91, 90, −85, 75, −63, 46, −28, 7, 13, −32, 50, −65, 78, −86,        90, −90, 85, −77, 64, −48, 29, −10}    - 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The techniques for encoding/decoding can be implemented by one or moreprocessors that may execute computer software having computer-readableinstructions that may be physically stored in one or morecomputer-readable media (e.g., a hard disk drive). For example, FIG. 7shows a computer system 700 suitable for implementing certainembodiments of the disclosed subject matter.

The computer software can be coded using any suitable machine code orcomputer language, that may be subject to assembly, compilation,linking, or like mechanisms to create code comprising instructions thatcan be executed directly, or through interpretation, micro-codeexecution, and the like, by computer central processing units (CPUs),Graphics Processing Units (GPUs), and the like.

The instructions can be executed on various types of computers orcomponents thereof, including, for example, personal computers, tabletcomputers, servers, smartphones, gaming devices, internet of thingsdevices, and the like.

The components shown in FIG. 7 for computer system 700 are exemplary innature and are not intended to suggest any limitation as to the scope ofuse or functionality of the computer software implementing embodimentsof the present disclosure. Neither should the configuration ofcomponents be interpreted as having any dependency or requirementrelating to any one or combination of components illustrated in theexemplary embodiment of a computer system 700.

Computer system 700 may include certain human interface input devices.Such a human interface input device may be responsive to input by one ormore human users through, for example, tactile input (such as:keystrokes, swipes, data glove movements), audio input (such as: voice,clapping), visual input (such as: gestures), olfactory input (notdepicted). The human interface devices can also be used to capturecertain media not necessarily directly related to conscious input by ahuman, such as audio (such as: speech, music, ambient sound), images(such as: scanned images, photographic images obtain from a still imagecamera), video (such as two-dimensional video, three-dimensional videoincluding stereoscopic video).

Input human interface devices may include one or more of (only one ofeach depicted): keyboard 701, mouse 702, trackpad 703, touch screen 710,data-glove 704, joystick 705, microphone 706, scanner 707, camera 708.

Computer system 700 may also include certain human interface outputdevices. Such human interface output devices may be stimulating thesenses of one or more human users through, for example, tactile output,sound, light, and smell/taste. Such human interface output devices mayinclude tactile output devices (for example tactile feedback by thetouch-screen 710, data-glove 704, or joystick 705, but there can also betactile feedback devices that do not serve as input devices), audiooutput devices (such as: speakers 709, headphones (not depicted)),visual output devices (such as screens 710 to include CRT screens, LCDscreens, plasma screens, OLED screens, each with or without touch-screeninput capability, each with or without tactile feedback capability someof which may be capable to output two dimensional visual output or morethan three dimensional output through means such as stereographicoutput; virtual-reality glasses (not depicted), holographic displays andsmoke tanks (not depicted)), and printers (not depicted).

Computer system 700 can also include human accessible storage devicesand their associated media such as optical media including CD/DVD ROM/RW720 with CD/DVD or the like media 721, thumb-drive 722, removable harddrive or solid state drive 723, legacy magnetic media such as tape andfloppy disc (not depicted), specialized ROM/ASIC/PLD based devices suchas security dongles (not depicted), and the like.

Those skilled in the art should also understand that term “computerreadable media” or “computer readable medium” as used in connection withthe presently disclosed subject matter does not encompass transmissionmedia, carrier waves, or other transitory signals.

Computer system 700 can also include interface to one or morecommunication networks. Networks can for example be wireless, wireline,optical. Networks can further be local, wide-area, metropolitan,vehicular and industrial, real-time, delay-tolerant, and so on. Examplesof networks include local area networks such as Ethernet, wireless LANs,cellular networks to include GSM, 3G, 4G, 5G, LTE and the like, TVwireline or wireless wide area digital networks to include cable TV,satellite TV, and terrestrial broadcast TV, vehicular and industrial toinclude CANBus, and so forth. Certain networks commonly require externalnetwork interface adapters that attached to certain general purpose dataports or peripheral buses (749) (such as, for example USB ports of thecomputer system 700; others are commonly integrated into the core of thecomputer system 700 by attachment to a system bus as described below(for example Ethernet interface into a PC computer system or cellularnetwork interface into a smartphone computer system). Using any of thesenetworks, computer system 700 can communicate with other entities. Suchcommunication can be uni-directional, receive only (for example,broadcast TV), uni-directional send-only (for example CANbus to certainCANbus devices), or bi-directional, for example to other computersystems using local or wide area digital networks. Certain protocols andprotocol stacks can be used on each of those networks and networkinterfaces as described above.

Aforementioned human interface devices, human-accessible storagedevices, and network interfaces can be attached to a core 740 of thecomputer system 700.

The core 740 can include one or more Central Processing Units (CPU) 741,Graphics Processing Units (GPU) 742, specialized programmable processingunits in the form of Field Programmable Gate Areas (FPGA) 743, hardwareaccelerators for certain tasks 744, and so forth. These devices, alongwith Read-only memory (ROM) 745, Random-access memory 746, internal massstorage such as internal non-user accessible hard drives, SSDs, and thelike 747, may be connected through a system bus 748. In some computersystems, the system bus 748 can be accessible in the form of one or morephysical plugs to enable extensions by additional CPUs, GPU, and thelike. The peripheral devices can be attached either directly to thecore's system bus 748, or through a peripheral bus 749. Architecturesfor a peripheral bus include PCI, USB, and the like.

CPUs 741, GPUs 742, FPGAs 743, and accelerators 744 can execute certaininstructions that, in combination, can make up the aforementionedcomputer code. That computer code can be stored in ROM 745 or RAM 746.Transitional data can be also be stored in RAM 746, whereas permanentdata can be stored for example, in the internal mass storage 747. Faststorage and retrieve to any of the memory devices can be enabled throughthe use of cache memory, that can be closely associated with one or moreCPU 741, GPU 742, mass storage 747, ROM 745, RAM 746, and the like.

The computer readable media can have computer code thereon forperforming various computer-implemented operations. The media andcomputer code can be those specially designed and constructed for thepurposes of the present disclosure, or they can be of the kind wellknown and available to those having skill in the computer software arts.

As an example and not by way of limitation, the computer system havingarchitecture 700, and specifically the core 740 can providefunctionality as a result of processor(s) (including CPUs, GPUs, FPGA,accelerators, and the like) executing software embodied in one or moretangible, computer-readable media. Such computer-readable media can bemedia associated with user-accessible mass storage as introduced above,as well as certain storage of the core 740 that are of non-transitorynature, such as core-internal mass storage 747 or ROM 745. The softwareimplementing various embodiments of the present disclosure can be storedin such devices and executed by core 740. A computer-readable medium caninclude one or more memory devices or chips, according to particularneeds. The software can cause the core 740 and specifically theprocessors therein (including CPU, GPU, FPGA, and the like) to executeparticular processes or particular parts of particular processesdescribed herein, including defining data structures stored in RAM 746and modifying such data structures according to the processes defined bythe software. In addition or as an alternative, the computer system canprovide functionality as a result of logic hardwired or otherwiseembodied in a circuit (for example: accelerator 744), which can operatein place of or together with software to execute particular processes orparticular parts of particular processes described herein. Reference tosoftware can encompass logic, and vice versa, where appropriate.Reference to a computer-readable media can encompass a circuit (such asan integrated circuit (IC)) storing software for execution, a circuitembodying logic for execution, or both, where appropriate. The presentdisclosure encompasses any suitable combination of hardware andsoftware.

While this disclosure has described several exemplary embodiments, thereare alterations, permutations, and various substitute equivalents, whichfall within the scope of the disclosure. It will thus be appreciatedthat those skilled in the art will be able to devise numerous systemsand methods which, although not explicitly shown or described herein,embody the principles of the disclosure and are thus within the spiritand scope thereof.

The invention claimed is:
 1. A method of decoding, the methodcomprising: receiving a video bitstream comprising a compressed datablock, wherein a discrete sine transform (DST) is to be applied as atransform core matrix to the compressed data block; determining whetherto use a first transform core matrix that is a 16-point, 8-bit DST-7primary transform core matrix, wherein the first primary transform corematrix is constructed using a set of sixteen integers {a, b, c, d, e, f,g, h, i, j, k, l, m, n, o, p}, wherein a value of {b} equals 17, a valueof {c} equals 25, a value of {d} equals 33, a value of {h} equals 62, avalue of {k} equals 77, a value of {l} equals 81 which is a sum of {a}and {j}, a value of {n} equals 87 which is a sum of {c} and {h};deriving the 16-point, 8-bit DST-7 primary transform core matrix byreordering the set of sixteen integers and determining theircorresponding signs; and reconstructing the compressed data block usingthe derived 16-point 8-bit DST-7 primary transform core matrix.
 2. Themethod of claim 1, further comprising: based on determining not use thefirst transform core matrix, deriving a second primary transform corematrix based on the set of sixteen integers, and reconstructing thecompressed data block using the derived second primary transform corematrix.
 3. An apparatus for decoding compressed video or image data, theapparatus comprising: at least one memory configured to store computerprogram code; and at least one processor configured to access the atleast one memory and operate according to the computer program code, thecomputer program code comprising: receiving code configured to cause theat least one processor to receive a video bitstream comprising acompressed data block, wherein a discrete sine transform (DST) is to beapplied as a transform core matrix to the compressed data block;determining code configured to cause the at least one processor todetermine whether to use a first transform core matrix that is a16-point, 8-bit DST-7 primary transform core matrix, wherein the firstprimary transform core matrix is constructed using a set of sixteenintegers {a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p} wherein avalue of {b} equals 17, a value of {c} equals 25, a value of {d} equals33, a value of {h} equals 62, a value of {k} equals 77, a value of {l}equals 81 which is a sum of {a} and {j}, a value of {n} equals 87 whichis a sum of {c} and {h}; deriving code configured to cause the at leastone processor to derive the 16-point, 8-bit DST-7 primary transform corematrix by reordering the set of sixteen integers and determining theircorresponding signs; and reconstructing code configured to cause the atleast one processor to reconstruct the compressed data block using thederived 16-point 8-bit DST-7 primary transform core matrix.
 4. Theapparatus of claim 3, wherein the computer program code furthercomprises: third applying code configured to cause the at least oneprocessor to, based on determining not use the first transform corematrix, derive a second primary transform core matrix based on the setof sixteen integers, and reconstruct the compressed data block using thederived second primary transform core matrix.
 5. A non-transitorycomputer-readable storage medium storing instructions that cause one ormore processors to: receive a video bitstream comprising a compresseddata block, wherein a discrete sine transform (DST) is to be applied asa transform core matrix to the compressed data block; determine whetherto use a first transform core matrix that is a 16-point, 8-bit DST-7primary transform core matrix, wherein the first primary transform corematrix is constructed using a set of sixteen integers {a, b, c, d, e, f,g, h, i, j, k, l, m, n, o, p} wherein a value of {b} equals 17, a valueof {c} equals 25, a value of {d} equals 33, a value of {h} equals 62, avalue of {k} equals 77, a value of {l} equals 81 which is a sum of {a}and {j}, a value of {n} equals 87 which is a sum of {c} and {h};deriving the 16-point, 8-bit DST-7 primary transform core matrix byreordering the set of sixteen integers and determining theircorresponding signs; and reconstruct the compressed data block using thederived 16-point 8-bit DST-7 primary transform core matrix.
 6. Thenon-transitory computer-readable storage medium of claim 5, wherein theinstructions cause the one or more processors to, based on determiningnot use the first transform core matrix, derive a second primarytransform core matrix based on the set of sixteen integers, andreconstruct the compressed data block using the derived second primarytransform core matrix.
 7. The method according to claim 1, wherein thecorresponding signs of each of the set of sixteen integers is positive.8. The method according to claim 1, wherein the 16-point, 8-bit DST-7primary transform core matrix comprises M rows and N columns, andwherein M is equal to N.
 9. The method according to claim 8, wherein afirst row of the M rows comprises the set of sixteen integers.
 10. Themethod according to claim 9, wherein a second row of the M rowscomprises a second set of sixteen integers at least one of whichcomprising a negative corresponding sign.
 11. The method according toclaim 1, wherein the first primary transform core matrix is a VersatileVideo Coding (VVC) standard transform.
 12. The apparatus according toclaim 3, wherein the corresponding signs of each of the set of sixteenintegers is positive.
 13. The apparatus according to claim 3, whereinthe 16-point, 8-bit DST-7 primary transform core matrix comprises M rowsand N columns, and wherein M is equal to N.
 14. The apparatus accordingto claim 13, wherein a first row of the M rows comprises the set ofsixteen integers.
 15. The apparatus according to claim 14, wherein asecond row of the M rows comprises a second set of sixteen integers atleast one of which comprising a negative corresponding sign.
 16. Theapparatus according to claim 3, wherein the first primary transform corematrix is a Versatile Video Coding (VVC) standard transform.
 17. Thenon-transitory computer-readable storage medium of claim 5, wherein thecorresponding signs of each of the set of sixteen integers is positive.18. The non-transitory computer-readable storage medium of claim 5,wherein the 16-point, 8-bit DST-7 primary transform core matrixcomprises M rows and N columns, and wherein M is equal to N.
 19. Thenon-transitory computer-readable storage medium of claim 18, wherein afirst row of the M rows comprises the set of sixteen integers.
 20. Thenon-transitory computer-readable storage medium of claim 19, wherein asecond row of the M rows comprises a second set of sixteen integers atleast one of which comprising a negative corresponding sign.